Comparison principles for nonlinear potential theories and PDEs with fiberegularity and sufficient monotonicity

We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets $\mathcal{F} \subset \mathcal{J}^2(X)$ of the 2-jets on $X \subset \mathbb{R}^n$ open) and subsolutions of degenerate elliptic and parabolic PDEs of the form $F(x,u,Du,D^2u) = 0$. We will implement the monotonicity-duality method begun by Harvey and Lawson in 2009 (in the pure second order constant coefficient case) for proving comparison principles for potential theories where $\mathcal{F}$ has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators $F$ which are compatible with $\mathcal{F}$ in a precise sense for which the correspondence principle holds. We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary. Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Example operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of G\r{a}rding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold.

for potential theories where F has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators F which are compatible with F in a precise sense for which the correspondence principle holds.
We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary.
Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Examples operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov In this work, we continue an investigation into the validity of the comparison principle u w on ∂Ω ⇒ u w in Ω (1.1) on bounded domains Ω in Euclidean spaces R n . We will operate in the two seemingly distinct frameworks of general second order nonlinear potential theories and of general second order fully nonlinear PDEs, where the formulations of comparison (1.1) in the two frameworks will soon be made precise. Comparison is of interest in both frameworks since, as is well known, it implies uniqueness of solutions to the natural Dirichlet problem in both frameworks (in the presence of some mild form of ellipticity). Moreover, comparison (1.1) together with suitable strict boundary convexity (that ensures the existence of needed barriers) leads to existence of solutions to the Dirichlet problem by Perron's method. In both frameworks we will also treat the following variant of the comparison principle u w on ∂ − Ω ⇒ u w in Ω, (1.2) where ∂ − Ω ∂Ω is a proper subset of the boundary. The version (1.1) "sees" the entire boundary and will hold under weak ellipticity assumptions and hence we will refer to it as the elliptic version of comparison. On the other hand, the version (1.2) which sees only a "reduced boundary" will be refered to as the parabolic version of comparison since it holds (for example) under weak parabolicity assumptions. While both versions of comparison are seemingly different in the two frameworks, we will connect the two frameworks for both versions by something called the correspondence principle which gives precise conditions of compatibility for which comparison in the two frameworks is equivalent. This is important for many reasons. A given second order potential theory on an open subset X ⊂ R n is determined by a constraint set F which is closed subset of J 2 (X) (the space of 2-jets on X) and which identifies a class of F -subharmonic functions on X, while a PDE on X is an equation of the form F (x, J) = 0 determined by an operator F acting on (x, J) ∈ J 2 (X). There may be many differential operators F organized about F which are compatible with the constraint set in the sense F = {(x, J) ∈ J 2 (X) : F (x, J) 0} and F = {(x, J) ∈ J 2 (X) : F (x, J) = 0}.
Hence potential-theoretic comparison for F gives operator-theoretic comparison for every operator F compatible with F . This is just one instance of the productive interplay between potential theory and operator theory. See the survey paper [15] and the preface of the monograph [7] for a more complete discussion of this interplay. We will have more to say on the origins and development of this program below, after presenting the two formulations of comparison and the main results obtained here, which will allow us to clearly underline what is new in this paper.
We now describe comparison in the first (potential theoretic) framework. Here one asks when does comparison (1.1) hold on Ω for each pair u ∈ USC(Ω) and w ∈ LSC(Ω) which are respectively F −subharmonic and F −superharmonic functions on Ω ⋐ X where F ⊂ J 2 (X) := X × J 2 = X × R × R n × S(n), X ⊂ R n open, is a subequation (constraint set) on X in the space of 2-jets. The precise definitions of subequations F and their associated subharmonics are given in Definitions 2.1 and 2.3, respectively. We note here only that F is closed and satisfies certain natural (monotonicity and topological) axioms which ensure that the potential theory determined by F (the associated F subharmonics) is meaningful and rich where the subharmonics on Ω satisfy the differential inclusion J 2,+ x u ⊂ F x := {J ∈ J 2 : (x, J) ∈ F }, ∀ x ∈ Ω in the viscosity sense where J 2,+ x u is the set of upper test jets for u in x. A general comparison principle is presented in Theorem 5.2 in this potential theoretic setting and the proof makes use of the monotonicity-duality method that was initiated in the constant coefficient pure second order case in Harvey-Lawson [9]. To use the method, we require three additional assumptions. First, the constraint set F much be sufficiently monotone in the sense that there exists a constant coefficient subequation M J 2 which is a monotonicity cone subequation for F ; that is, in addition to being a subequation it is also a convex cone with vertex at the origin such one has the monotonicity property We also require that the constraint set F is fiberegular in the sense that the fiber map is continuous with respect to the Euclidian metric on X and the Hausdorff metric on the closed subsets K (J 2 ) of J 2 . This notion was introduced in [5] in the variable coefficient pure second order case and then was extended to the gradient-free case in [6]. Finally, for the elliptic version of comparison (1.1) on Ω, we require that the monotonicity cone M admits a function ψ ∈ C 2 (Ω) ∩ USC(Ω) which is strictly M-subharmonic on Ω. For the the parabolic version of comparison (1.2) on Ω, we also require that ψ blows up on the complement of the reduced boundary in the sense that ψ ≡ −∞ on ∂Ω \ ∂ − Ω. (1.4) The utility of the General Comparison Theorem 5.2 is greatly enhanced by exploiting by the detailed study of monotonicity cone subequations in [7], which we briefly review. There is a three parameter fundamental family of monotonicity cone subequations (see Definition  where F ∈ C(G) with either G = J 2 (X) (the unconstrained case) or G J 2 (X) is a subequation (the constrained case). The proper ellipticity means the following monotonicity property: for each x ∈ X and each (r, p, A) ∈ G x one has F (x, r, p, A) F (x, r + s, p, A + P ) ∀ s 0 in R and ∀ P 0 in S(n). (1.6) Notice that one of the subequation axioms on G is the monotonicity property that for each x ∈ X one has G x + M 0 ⊂ G x where M 0 := {(r, p, A) : s 0 in R and ∀ P 0 in S(n)}, which is needed in (1.6). We will refer to (F, G) as a proper elliptic pair (see Definition 7.1). Notice also that proper ellipticity is the familiar (opposing) monotonicity in solution variable r and the Hessian variable A, which we do not necessarily assume globally on all of J 2 (X) (but do in the unconstrained case). In general, a given operator F must be retricted to a subequation in order to have the minimal monotonicity needed (1.6). For example, the Monge-Ampère operator F (x, r, p, A) := det(A) must be restricted to the constraint G := {(r, p, A) ∈ J 2 : A ∈ S(n) : A 0} in order to be increasing in A. In this case, the G-admissible subsolutions are convex and one uses only convex lower test functions in the definition of G-admissible subsolutions.
We will deduce a general operator theoretic comparison Theorem 8.1 from the potential theoretic comparison Theorem 5.2 by way of the aforementioned Correspondence Principle of Theorem 7.4. This correspondence consists of the two equivalences: for every u ∈ USC(X) u is F -subharmonic on X ⇔ u is a G-admissible subsolution of F (J 2 u) = 0 on X and u is F -superharmonic on X ⇔ u is a G-admissible supersolution of F (J 2 u) = 0, where (F, G) is a proper elliptic pair and F is the constraint set defined by the correspondence relation F = {(x, J) ∈ G : F (x, J) 0}. (1.7) The correspondence principle holds provided that F is itself a subequation and provided that one has compatibility int F = {(x, J) ∈ G : F (x, J) > 0}, which for subequations F defined by (1.7) is equivalent to Now, F is indeed a subequation by Theorem 7.8 if the following three hypotheses are satisfied: (i) G is fiberegular in the sense (1.3) (with F replaced by G), (ii) (F, G) is M-monotone for some monotonicity cone subequation and (iii) (F, G) satisfies the following regularity condition: for some fixed J 0 ∈ int M, given Ω ⋐ X and η > 0, there exists δ = δ(η, Ω) > 0 such that Not only is F a subequation (and hence the correspondence principle holds), F is also fiberegular and M-monotone. Hence one will have both operator theoretic and potential theoretic comparison in both versions (1.1) and (1.2) on any domain Ω which admits a C 2 strictly M-subharmonic function (which also satisfies (1.4) in the parabolic version).
Having described the main general comparison theorems in both frameworks, we now place them in context to indicate what is new in the paper. In the important special case of constant coefficient subequations F ⊂ J 2 and constant coefficient operators F ∈ C(G) with constant coefficient admissibility constraint G ⊆ J 2 , the entire program (and much more) has been developed in the research monograph [7]. In particular, it is there in Chapter 11 that the important bridge of the Correspondence Principle was refined into its present form. In the current paper, variable coefficients require the fiberegularity conditions like (1.3) in order to overcome the essential difficulty that the sup-convolutions used to approximate upper semicontinuous F -subharmonics with quasi-convex 1 functions do not preserve the property of being F -subharmonic in the variable coefficient case. Here fiberegularity ensures what we call the uniform translation property for subharmonics which roughly states that (see Theorem 3.3): if u ∈ F (Ω), then there are small C 2 strictly F -subharmonic perturbations of all small translates of u which belong to F (Ω δ ), where Ω δ := {x ∈ Ω : d(x, ∂Ω) > δ}. 1 We have adopted the term quasi-convex which is consistent with the use of quasi-plurisubharmonic function in several complex variables. Quasi-convex functions are often referred to as semiconvex functions, although this term is a bit misleading. They are functions whose Hessian (in the viscosity sense) is locally bounded from below.
The formulation of the fiberegularity condition on F and the proof that it implies the uniform translation property was done first in the variable coefficient pure second order case where in [5] and then extended to the variable coefficient gradient-free second order case where in [6]. However, in these papers, the term fiberegularity was not used. The term fiberegularity was coined in the production of the survey paper [15]. The terminology of [5] and [6] borrowed much from the fundamental paper of Krylov [19] on the general notion of ellipticity. More importantly, the form of the correspondence principle in [5] and [6] was more rudimentary than the form described above. The present paper adds additional results and refinements to the variable coefficient pure second order and gradient-free situations of [5] and [6], which heavily benefit from the investigation of the constant coefficient case in [7].
This brings us to the main issue of this paper, which is establishing comparison in both elliptic and parabolic versions (1.1) and (1.2) for variable coefficient potential theories and PDEs with directionality in the gradient variables. Definition 1.1. A closed convex cone D ⊆ R n (possibly all of R n ) with vertex at the origin and int D = ∅ will be called a directional cone. We say that a subequation F ⊂ J 2 (X) on an open subset X satisfies the directionality condition (with respect to D) if Notice that when D = R n , the D-directionality just means that F is gradient-free. Hence we will be particularly interested in situations where there is D-directionality of the subequation F with a directional cone D R n , in order to extend what is known from the gradient-free case in [6]. Similarly, for a given proper elliptic pair (F, G) we will be particularly interested in situations in which G satisfies (D) with D R n and the natural property of directionality in the gradient variables F (x, r, p + q, A) F (x, r, p, A) for each (r, p, A) ∈ G x , q ∈ D, x ∈ X.
Some interesting directional cones D R n are given in Example 12.33 of [7] and we recall two of them here: We now describe two example applications of the general comparison theorems to interesting fully nonlinear PDEs with directionality in the gradient from Section 8; an elliptic example and a parabolic example. The elliptic example concerns equations that arise the theory of optimal transport and is the following example. Example 1.2 (Example 8.2 (Optimal transport)). The equation describes the optimal transport plan from a source density f to a target density g. In Proposition 8.3 we will prove the elliptic version of comparison under the hypotheses that f, g ∈ C(Ω) and are nonnegative (1.11) and that g is D-directional with respect to some directional cone D R n ; that is, , for each p, q ∈ D. (1.12) We also require some measure of strict directionality in the sense that there exists q ∈ int D and a modulus of continuity ω : (0, ∞) → (0, ∞) (satisfying ω(0 + ) = 0) such that g(p + ηq) ≥ g(p) + ω(η), for each p, q ∈ D and each η > 0. (1.13) The natural operator F associated to (1.10) is defined F (x, r, p, A) := g(p)det(A) − f (x) and is proper elliptic when restricted to A 0 in S(n). The compatible subequation F with fibers is fiberegular and M-monotone for As shown in [7], these cones admit C 2 strictly M subharmonics on all bounded domains so one has potential theoretic comparison for F as well as operator theoretic comparison for G-admissible subsolutions, supersolutions of (1.10) with G = M(D, P).
The parabolic example that we describe is a prototype of a fully nonlinear PDE which is weakly parabolic in the sense of Krylov and also indicates the utility of half-space cones (in the gradient variable) (1.8) in this parabolic context. Example 1.3 (Example 8.4 (Krylov's parabolic Monge-Ampère operator)). In [18], the following nonlinear parabolic equation is considered where Ω ⋐ R n is open and T > 0. The reduced boundary of the parabolic cylinder X is which is the usual parabolic boundary of X. In Proposition 8.5, for arbitrary bounded parabolic cylinders X and f ∈ C(X) nonnegative, we prove the parabolic version of comparison for G-admissible subsolutions, supersolutions u, v of the equation (1.14), where the admissibility constraint is the natural constant coefficient subequation with constant fibers where A n ∈ S(n) is the upper-left n×n submatrix of A. This is because the compatible subequation F with fibers is fiberegular and M(D n , P n )-monotone, where every parabolic cylinder X admits a strictly C 2 and M(D n , P n )-subharmonic function ψ which satisfies and hence the parabolic version of Theorem 8.1 applies.
Many additional examples of fully nonlinear operators with directionality in the gradient variables can be constructed from Dirichlet-Gårding polynomials on the vector space V = R n , which are homogeneous polynomials g of degree m are hyperbolic with respect to the direction q ∈ R n in the sense that the one-variable polynomial t → g(tq + p) has exaclty m real roots for each p ∈ R n .
(1.16) See Definition 8.6 and the brief discussion which follows on concerning elements of Gårding's theory of hyperbolic polynomials. The key point is that one represent the first order operator determined by g as a generalized Monge-Ampère operator in the sense that g(p) = λ g 1 (p) · · · λ g m (p). For k = 1, . . . , m, the factor λ g k (p) is the k-th Gårding q-eigenvalue of g, which is just the negative of the k-th root in (1.16) (reordered so that λ g 1 (p) λ g 2 (p) · · · λ g m (p)). There is always a natural monotonicty cone Γ (the (closed) Gårding cone) for a hyperbolic polynomial g, which in the case V = R n is a directional cone D.
In order to illustrate the construction above, in Example 8.7 we discuss the polynomial defined for p = (p 1 , p 2 ) ∈ R 2 by g(p 1 , p 2 ) := p 2 1 − p 2 2 = λ g 1 (p)λ g 2 (p) = (p 1 − |p 2 |)(p 1 + |p 2 |), which determines the pure first order fully nonlinear equation The associated directional cone is In Proposition 8.8 we prove a parabolic version of comparison on rectangular domains Ω ⊂ R 2 for G-admissible subsolutions, supersolutions of (1.17) with respect to the natural admissibility constraint which is also the monotonicity cone subequation for comparison. As a final example of a fully nonlinear operator with directionality in the gradient, we will discuss the following interesting operator. Example 1.4 (Example 8.9 Perturbed Monge-Ampère operators with directionality). On a bounded domain Ω ⊂ R n , consider the operator defined by with f ∈ U C(Ω; [0, +∞)) and with M ∈ U C(Ω × R n ; S(n)) of the form with P ∈ U C(Ω; P) and b ∈ U C(Ω; R n ) such that there exists a unit vector ν ∈ R n such that b(x), ν 0 for each x ∈ Ω.
Such operators with M = M (x) have been proposed by Krylov as interesting test cases for probabilistic and analytic methods. Our interest in this example is two fold. On the one hand, we can prove the elliptic version of comparison using our methods with a natural directional cone See the discussion in Example 8.9. On the other hand, we show in Proposition 8.10 standard viscosity structural conditions fail for F , and hence this example shows that our methods can provide comparison in non standard cases with directionality in the gradient (as was already known in the pure second order and gradient-free settings from [5] and [6]).
While our main focus here is on the (strict) directionality with respect to first order terms, we stress that our theory allows us to treat operators that are parabolic in a broad sense. For instance, pure linear second-order operators tr(BD 2 u) are classically considered to be parabolic provided that B ≥ 0 and det B = 0. Hence, there is a natural nontrivial monotonicity cone associated to the (restricted) subequation F = {A : tr(BA) ≥ 0} which is M = F . Therefore, any operator F that can be paired with a subequation with such monotonicity cone M can be regarded as parabolic, and specific comparison principles on suitable restricted boundaries can be easily deduced.
Fundamental to the entire project we discuss here is the groundbreaking paper of Harvey and Lawson [9] which examined the potential theoretic comparison as well as existence via Perron's method in the constant coefficient case pure second order case for potential theories F ⊂ S(n).
No correspondence principle is found there, as the focus was on the potential theory side. This is because in the geometric situations of interest to them, often there are no natural operators associated to geometric potential theories of interest. It was in [9] that Krylov's fundamental insight to associate constraint sets which encode the natural notion of ellipticity for differential operators takes shape and is encoded by them in the language of general (nonlinear) potential theories. Moreover, in [9] the natural notion of duality (the Dirichlet dual) is formalized. It is implicit in [19], but made explicit in [9] and used elegantly to clarify the notion of supersolutions in constrained cases and as a crucial ingredient of their monotonicity-duality method for proving comparison. This method is presented in Section 5, which for the first time incorporates the parabolic version into the crucial Zero Maximum Principle (ZMP) in Theorem 5.1 (note that this is new also for pure second order and gradient-free settings).
Two remarks on the crucial fiberegularity condition (the continuity of the fiber map Θ of (1.3)) are in order. First, the recent interesting work of Brustad [3] in the pure second order setting (operators without first and zero-th order terms), introduces a regularity property for the fiber map Θ which is weaker than the fiberegularity used here. A concise discussion this weaker condition is given in the introduction of [3] which aims to incorporate the best features of fiberegulairty and standard viscosity structual conditions in this case. Second, the recent important paper of Harvey and Lawson [14], which studies the Dirichlet problem for inhomogeneous equations on manifolds X under the assumptions that (F, G) is an M-monotone compatible operator-subequation pair for which the operator is tame. In the constant coefficient case on R n this condition requires that for every s, λ > 0 there exists c(s, λ) > 0 such that F (J + (r, 0, P )) − F (J) c(s, λ), ∀ J ∈ G and P λI in S(n). (1.20) This property, which not comparable to the fiberegularity of F := {J ∈ G : F (J) 0}, plays the same role as fiberegularity in this inhomogeneous setting. We conclude this introduction with a brief description of the contents. Part 1 of the paper (Sections 2 -6) concerns the potential theoretic setting, including the elliptic and parabolic versions of comparison by the monotonicity-duality method in the presence of fiberegularity. Section 6 also gives some new characterizations of dual cone subharmonics that play a crucial role in comparison by way of the (ZMP). Part 2 of the paper is Section 7 which builds the bridge between the potential theoretic framework and the operator theoretic framework by way of the correspondence principle. Part 3 of the paper treats comparison in the operator theoretic framework and is highlighted by the examples mentioned above.
In addition there are three appendices. Appendix A contains many new auxilliary technical results needed to complete the proof of of Theorem 7.11 which proves that given an M-monotone pair (F, G) the natural constraint set F defined by the correspondence relation is a fiberegular M-monotone subequation if G is fiberegular. This theorem plays an important role in the general PDE comparison principle Theorem 8.1. Appendix B collects some known results which are fundamental for the potential theoretic methods and is included for the convenience of the reader. Appendix C recalls some elementary facts about the Hausdorff distance which are used in the discussion of fiberegularity in Section 3.

Background notions from nonlinear potential theory
In this section, we give a brief review of some key notions and fundamental results in the theory of F -subharmonic functions defined by a subequation constraint set F .

Subequation constraint sets and their subharmonics.
Suppose that X is an open subset of R n with 2-jet space denoted by J 2 (X) = X × J 2 = X × (R × R n × S(n)). A good definition of a constraint set with a robust potential theory was given in [10] (also for manifolds).
Definition 2.1 (Subequations). A set F ⊂ J 2 (X) is called a subequation (constraint set) if (P) F satisfies the positivity condition (fiberwise); that is, for each x ∈ X (r, p, A) ∈ F x ⇒ (r, p, A + P ) ∈ F x , ∀ P 0 in S(n).
(T) F satisfies three conditions of topological stability 2 : Notice that by property (T3) we can write without ambiguity int F x for the subset of J 2 , which can be calculated in two ways. The conditions (P), (T) and (N) have various (important) implications for the potential theory determined by F . Some of these will be mentioned below (see the brief discussion following Definition 2.3). In addition, the conditions (P) and (N) are monotonicity properties; monotonicity plays a central and unifying role as will be discussed in Subsection 2.3. The role of property (T) is clarified by the notion of duality; another fundamental concept that will be discussed in Subsection 2.2. For now, notice that by property (T1), F is closed in J 2 (X) and each fiber F x is closed in J 2 by (T2). In addition, the interesting case is when each fiber F x is not all of J 2 , which we almost always assume. Also notice that in the constant coefficient pure second order case where the (reduced) subequation 3 can be identified with a subset F ⊂ S(n), property (N) is automatic and property (T) reduces to (T1) F = int F , which is implied by (P) for F closed. Hence in this case, subequations F ⊂ S(n) are closed sets simply satisfying (P). Additional considerations on property (T) will be discussed in Appendix A.
Next we recall the notion of F -subharmonicity for a given subequation F ⊂ J 2 (X). There are two different natural formulations for differing degrees of regularity. The first is the classical formulation.
with the accompanying notion of being strictly F -subharmonic if For merely upper semicontinuous functions u ∈ USC(X) with values in [−∞, +∞), one replaces the 2-jet J 2 x u with the set of C 2 upper test jets J 2,+ x u := {J 2 x ϕ : ϕ is C 2 near x, u ϕ near x with equality at x}, (2.3) thus arriving at the following viscosity formulation.
We denote by F (X) the set of all F -subharmonics on X.
We now recall some of the implications that properties (P), (T) and (N) have on an F -potential theory. Property (P) ensures that Definition 2.3 is meaningful since for each u ∈ USC(X) and for each x 0 ∈ X one has property (P) for the upper test jets (r, p, A) ∈ J 2,+ x0 u ⇒ (r, p, A + P ) ∈ J 2,+ x0 u, for each P 0 in S(n). (2.5) Indeed, given an upper test jet J 2 x0 ϕ = (r, p, A) with ϕ a C 2 function near x 0 and satisfying u ϕ near x 0 with equality at x 0 then, for each P 0, the quadratic perturbation ϕ(·) := ϕ(·) + 1 2 P (· − x 0 ), (· − x 0 ) determines an upper test jet J 2 x0 ϕ = (r, p, A + P ). Property (P) is also crucial for C 2 -coherence, meaning classical F -subharmonics are F -subharmonics in the sense (2.4), since for u which is C 2 near x, one has where P = {P ∈ S(n) : P 0}. Next note that property (T) insures the local existence of strict classical F -subharmonics at points x ∈ X for which F x is non-empty. One simply takes the quadratic polynomial whose 2-jet at x is J ∈ int(F x ). Finally, property (N) eliminates obvious counterexamples to comparison. The simplest counterexample is provided by the constraint set F ⊂ J 2 (R) in dimension one associated to the equation u ′′ − u = 0, which is defined by F := {(r, p, A) ∈ R 3 : A − r 0}.

2.2.
Duality and superharmonics. The next fundamental concept is duality, a notion first introduced in the pure second order coefficient case in [9]. Definition 2.4 (Duality for constraint sets). For a given subequation F ⊂ J 2 (X) the Dirichlet dual of F is the set F ⊂ J 2 (X) given by 4 With the help of property (T), the dual can be calculated fiberwise This is a true duality in the sense that one can show the following two facts: Here and below, c denotes the set theoretic complement of a subset.
Additional (and useful) properties of the dual can be found in Propositions 3.2 and 3.4 of [7]. These properties include the behavior of the dual with respect to inclusions, intersections and fiberwise sums: This last formula, when combined with the monotonicity discussed below, will lead to the fundamental formula (2.19) for the monotonicity-duality method.
Another importance of duality is that it can be used to reformulate the notion of F -superharmonic functions in terms of dual subharmonic functions. This will have important implications for the correct definition of supersolutions to a degenerate elliptic PDE F (J 2 u) = 0 in the presence of admissibility constraints. See Subsection 7.1 for this discussion.
The natural notion of w ∈ LSC(X) being F -superharmonic using lower test jets is which by duality and property (T) is equivalent to −w ∈ USC(X) satisfying This fundamental notion appears in various guises. It is a useful and unifying concept. One says that a subequation F is M-monotone for some subset M ⊂ J 2 (X) if For simplicity, we will restrict attention to (constant coefficient) monotonicity cones; that is, monotonicity sets M for F which have constant fibers which are closed cones with vertex at the origin. First and foremost, the properties (P) and (N) are monotonicity properties. Property (P) for subequations F corresponds to degenerate elliptic operators F and properties (P) and (N) together correspond to proper elliptic operators. Note that (P) plus (N) can be expressed as the single monotonicity property Hence M 0 will be referred to as the minimal monotonicity cone in J 2 . However, it is important to remember that M 0 ⊂ J 2 is not a subequation since it has empty interior so that property (T) fails. A monotonicity cone which is also a subequation will be called a monotonicity cone subequation.
Combined with duality and fiberegularity (defined in Section 3), one has a very general, flexible and elegant geometrical approach to comparison when a subequation F admits a constant coefficient monotonicity cone subequation M. We call this approach the monotonicity-duality method and it will be discussed in Section 5. One key point in the method is the following monotonicity-duality formula that combines monotonicity (2.15) and the duality formula on fiberwise sums (2.11): It is interesting to note that if a subequation F has a constant coefficient monotonicity cone subequation M then the fiberwise sum of F and its dual F yields a constant coefficient subequation M which is also a cone (dual to the monotonicity cone for F and F ). A detailed study of monotonicity cone subequations can be found in Chapters 4 and 5 of [7], including the construction of a fundamental family of monotonicity cones that is recalled below in (5.5)-(5.6). Monotonicity is also used to formulate reductions when certain jet variables are "silent" in the subequation constraint F . For example, one has and M(N , P) are fundamental constant coefficient (cone) subequations which can be identified with P ⊂ S(n) and Q := N × P ⊂ R × S(n). One can identify F with subsets of the reduced jet bundles X × S(n) and X × (R × S(n)), respectively, "forgetting about" the silent jet variables (see Chapter 10 of [7]). For a more extensive review of the monotonicity, see subsection 2.2 of [15].
Three important "reduced" examples are worth drawing special attention to. They are all monotonicity cone subequations and play a fundamental role in our method. We focus on characterizing their subharmonics and their dual subharmonics.
Example 2.5 (The convexity subequation). The convexity subequation is F = X × M(P) and reduces to X × P which has constant coefficients (each fiber is P) and for u ∈ USC(X) u is P-subharmonic ⇔ u is locally convex (away from any connected components where u ≡ −∞).
The convexity subequation has a so-called canonical operator F ∈ C(S(n), R) defined by the minimal eigenvalue F (A) := λ min (A), for which (2.20) The dual subequation F has constant fibers given by which is the subaffine subequation. The set P(X) of dual subharmonics agrees with SA(X) the set of subaffine functions defined as those functions u ∈ USC(X) which satisfy the subaffine property (comparison with affine functions): for every Ω ⋐ X one has u a on ∂Ω ⇒ u a on Ω, for every a affine.
The fact that P(X) = SA(X) is shown in [9]. The subaffine property for u is stronger than the maximum principle for u since constants are affine functions. It has the advantage over the maximum principle of being a local condition on u. This leads to the comparison principle for all pure second order constant coefficient subequations [9] and extends to variable coefficient subequations [5] using the notion of fiberegularity noted above.
Example 2.6 (The convexity-negativity subequation). The constant coefficient gradient-free subequation F = X × M(N , P) reduces to X × Q ⊂ X × (R × S(n)) whose (constant) fibers are The (reduced) dual subequation has (constant) fibers The set Q(X) of dual subharmonics agrees with SA + (X), the set of subaffine plus functions defined as those functions u ∈ USC(X) which satisfy the subaffine plus property: for every Ω ⋐ X one has u a on ∂Ω ⇒ u a on Ω, for every a affine with a |Ω 0.
(2.25) from which the Zero Maximum Principle (ZMP) of Theorem 5.1 for Q subharmonics follows immediately. The fact that Q(X) = SA + (X) is shown in [7] together with the additional equivalence This leads to the comparison principle by the monotonicity-duality method for all gradient free subequations with constant coefficients in [7] and extends to variable coefficient gradient-free subequations in [6], using the notion of fiberegularity.
The third example is many respects the focus of the present work, as it treats a sufficient monotonicity in the gradient variables for the monotonicity-duality method when the gradient variables are present. In this section, we will limit ourselves to characterizing the subharmonics, which is interesting in its own right. The characterization of the dual subharmonics will be done in section 6 in the general context of characterizing dual cone subharmonics.
Example 2.7 (The directionality subequation). Consider a directional cone D ⊂ R n as defined in Definition 1.1; that is, a closed convex cone with vertex at the origin and non empty interior int D. The directionality subequation is the constant coefficient pure first order subequation F = X × M(D) = X × (R × D × S(n)) reduces to X × D ⊂ X × R n whose (constant) fibers are the directional cone D. The (reduced) dual subequation has (constant) fibers given by the Dirichlet dual which is also a directional cone. Two examplesof directional cones were recalled in (1.8) and (1.9).
The following characterization of M(D) subharmonics is new.
Proposition 2.8 (Directionality subharmonics are increasing in polar directions). Suppose that D ⊂ R n is a directional cone with polar cone 5 The set of M(D)-subharmonics can be characterized as follows: Proof. Indeed, assume first that (2.29) holds. By Definition 2.3, we need to show that for each Therefore, by a Taylor expansion of ϕ and (2.29), Dividing by r > 0 and taking the limit To show the other implication, we need some machinery from nonsmooth and convex analysis. First, for an M(D)-subharmonic function u we consider the sequence of sup-convolutions u ε (see (4.4) below). We have that u ε is 1 ε -quasi-convex and decreases pointwise to u as ε ց 0. Moreover, for any Ω ⊂⊂ X, since M(D) has constant coefficients, u ε is M(D)-subharmonic on Ω for ε small enough and, by Alexandroff's theorem, u ε is almost everywhere twice differentiable, so for a.e. x ∈ Ω.
Note that for any such point, Du ε (x) represents the generalized subgradient ∂u ε (x) (see for example [4,Section 2]). In fact, for every x ∈ Ω, ∂u ε (x) is given by the convex hull of limit points of (converging) sequences Du ε (x n ), where x n → x (see [4, Theorem 2.5.1]). Since we can choose x n such that Du ε (x n ) ∈ D and D is a closed convex cone, we get that ∂u ε (x) ⊆ D for every x ∈ Ω, and therefore ∂u ε (x), q is a subset of nonnegative reals for every q ∈ D • and x ∈ Ω. Finally, if . Passing to the limit ε → 0 + yields the defired conclusion (2.29). 5 We follow the convention of [7] in calling D • defined by (2.28) the polar cone determined by the set D. Some call this set the dual cone and denote it by D * and then define the polar cone as −D * . Our choice avoids confusion with the (Dirichlet) dual cone (2.27).

Fiberegularity
In this section we discuss a fundamental notion which is crucial in the passage from constant coefficient subequations (and operators) to ones with variable coefficients. We begin with the definition.
is taken as the norm on J 2 where λ 1 (A) · · · λ n (A) are the (ordered) eigenvalues of A ∈ S(n). We will also make use of some standard facts concerning the Hausdorff distance in the proof of Proposition 3.2 below; these facts will be recalled in Appendix C for the reader's convenience.
This notion was first introduced in [5] in the special case F ⊂ X × S(n). We will also refer to Θ as a continuous proper ellipitc map since it takes values in the closed (non-empty and proper) subsets of J 2 satisfying properties (P) and (N).
Note that by the Heine-Cantor Theorem, fiberegularty is equivalent to the local uniform continuity of the fiber map Θ. Moreover, if F is M-monotone for some (constant coefficient) monotonicity cone subequation, fiberegularity has more useful equivalent formulations. (a) Θ is locally uniformly continuous, that is for each Ω ⋐ X and every η > 0 there exists δ = δ(η, Ω) > 0 such that (b) Θ is locally uniformly upper semicontinuous (in the sense of multivalued maps), that is for each Ω ⋐ X and every η > 0 there exists δ = δ(η, Ω) > 0 such that Moreover, the validity of this property for one fixed J 0 ∈ int M implies the validity of the property for each J 0 ∈ int M.
Formulation (c) is the most useful definition of fiberegularity for M-monotone subequations. In the pure second order and gradient-free cases there is a "canonical" reduced jet J 0 = I ∈ S(n) and J 0 = (−1, I) ∈ R × S(n), respectively. (a) implies (c) for every J 0 ∈ int M. By definition (C.4) we have, for I, K ⊂ J 2 , Fix now J 0 ∈ int M and η > 0; if Θ is uniformly continuous on Ω, then, for η ′ > 0 to be determined, there exists δ = δ(η ′ , Ω) such that Hence for x, y ∈ Ω with |x − y| < δ one has We want to show that for each J ∈ Θ(x), one has Using the decomposition (3.3), so that, by the M-monotonicity of Θ(y), one has (3.4) provided that (b) implies (a) This is a standard proof which does not require any monotonicity assumption. For hence, thanks to the first inclusion, for every J ∈ Θ(x) there exists J ′ ∈ Θ(y) such that J = J ′ + K for some K with |||K||| < η ′ . Therefore By the second inclusion in (3.6), one also has and thus by (3.2) Fiberegularity is crucial since it implies the uniform translation property for subharmonics. This property is the content of the following result, which roughly speaking states that: if u ∈ F (Ω), then there are small C 2 strictly F -subharmonic perturbations of all small translates of u which belong to F (Ω δ ), where Ω δ := {x ∈ Ω : d(x, ∂Ω) > δ}. Theorem 3.3 (Uniform translation property for subharmonics). Suppose that a subequation F is fiberegular and M-monotone on Ω ⋐ R n for some monotonicity cone subequation M. Suppose that M admits a strict approximator 6 where τ y u( · ) := u( · − y).
Proof. We are going to use the Definitional Comparison Lemma B.2 in order to adapt the method used in the proofs of the pure second-order and the gradient-free counterparts of this uniform translation property (see [5,Proposition 3.7(5)] and [6, Proposition 3.7(4)]). Fix J 0 ∈ int M and let δ = δ(η, Ω) be as in Proposition 3.2(c), with η > 0 to be determined. Consider Ω ′ ⋐ Ω δ and v ∈ C 2 (Ω ′ ) ∩ USC(Ω ′ ), strictly F -subharmonic on Ω ′ . In order to prove the subharmonicity of u y;θ (defined as in (3.7)) via the definitional comparison (cf. Remark B.3), it suffices to show that, for a suitable η, Fix θ > 0 and for y ∈ B δ consider the function v y;θ := τ −y v + θτ −y ψ, 6 The term strict approximator for ψ refers to the fact that this function generates an approximation from above of the M-subharmonic function which is identically zero. This is explained in the proof of Theorem 6.2 of [7].
defined on Ω ′ + y, which satisfies therefore, by the M-monotonicity of F , and using (3.9), Since ψ is a strict approximator for M on Ω, we know that there exists ρ(x) > 0 such that in order for (3.11) to hold for any Ω ′ ⋐ Ω δ . It is worth noting that the bound (3.12) is independent of δ, which does depend on η, and hence an η satisfying (3.12) can be chosen.
The uniform translation property of Theorem 3.3 will play a key role in the treatment of the variable coefficient setting, where one does not have translation invariance. In particular, it will be used to show that given a semicontinuous F -subharmonic function u there are quasi-convex approximations of u which remain F -subharmonic provided that F is fiberegular and M-monotone (see Theorem 4.2).
Remark 3.4. Concerning the additional hypothesis that the monotonicity cone subequation M admits a C 2 strict subharmonic, we note that in the pure second order and gradient-free cases (F ⊂ Ω × S(n) and F ⊂ Ω × (R × S(n)), one always has a quadratic strict approximator ψ. Thus Theorem 3.3 holds for all continuous coefficient F which are minimally monotone (with M = P ⊂ S(n) and M = Q = N × P ⊂ R × S(n) respectively). In the general M-monotone and fiberegular case this additional hypothesis will be essential in the proof of the so-called Zero Maximum Principle (ZMP) of Theorem 5.1 for the dual monotonicity cone M. The (ZMP) is a key ingredient in the monotonicty-duality method for proving comparison, as wil be discussed below in Section 5. Moreover, the (constant coefficient) monotonicity cone subequations which admit strict approxiamtors are well understood by the study made in [7] and will be recalled below in Theorem 5.3 and the discussion which precedes the theorem.
Fiberegularity of an M-monotone subequation has two additional consequences which are of use in treating existence by Perron's method. While we will not pursue existence here, we record the result for future use. A general property of uniformly continuous maps on some open subset Ω with boundary ∂Ω is the possibility to extend them to the boundary. One can prove that the M-monotonicity is preserved as well. Also, one can define in a natural way the dual fiber map of Θ by note that this is a pointwise (or fiberwise) definition, and that by a straightforward extension to variable coefficient subequations of the elementary properties of the Dirichlet dual collected in [9,Section 4], [10,Section 3] The following proposition, which extends [6, Proposition 3.6], collects these two properties. Proof. (a) We essentially reproduce the proof of [5, Proposition 3.5]. One extends Θ to x ∈ ∂Ω as a limit where {x k } k∈N ⊂ Ω is a sequence such that lim k→∞ x k = x and the limit in (3.13) is to be understood in the complete metric space (K(J 2 ), d H ). This limit exists since {x k } is Cauchy sequence and hence so is {Θ(x k )} by the uniform continuity of Θ. Moreover, this limit is independent of the choice of {x k }, and we have the extension of Θ to ∂Ω by performing this construction for each x ∈ ∂Ω. The resulting extension is uniformly continuous and each Θ(x) is closed by construction. It remains to show that the extension takes values in the set of M-monotone sets. First of all, each Θ(x) is non-empty because d H (Θ(x), ∅) = +∞ for all x ∈ Ω (by property Remark C.3) and hence Θ(x k ) → ∅. As for the M-monotonicity of the limit set Θ(x), note that by (3.2) it is easy to show that Θ(x) is the set of limits of all converging sequences and thus for eachĴ ∈ M by the M-monotonicity of each Θ(x k ); hence J +Ĵ ∈ Θ(x) for each J ∈ Θ(x) and eachĴ ∈ M.
Finally, to prove that Θ(x) is a proper subset of J 2 , it suffices to invoke Lemma C.4; indeed, arguing as above, this guarantees that Θ(x k ) → J 2 .
(b) We proceed as in the proof of [5, Proposition 3.5]. As we already noted, by the elementary properties of the Dirichlet dual (namely [7, Proposition 3.2, properties (2) and (6)]), one knows that Θ is M-monotone if and only if Θ is (note that this is a fiberwise property); then we only need to show that Θ is uniformly continuous on Ω. Since Θ is uniformly continuous on Ω, by Proposition 3.2(c), for η > 0, J 0 ∈ int M, and a suitable δ = δ(η, Ω, J 0 ) one has whenever x, y ∈ Ω are such that |x − y| < δ. Hence, by the above-mentioned elementary properties of the Dirichlet dual, one obtains thus proving the uniform continuity of Θ by exploiting the equivalent formulation of Proposition 3.2(c) again.
Remark 3.6. It is worth noting that this proof shows that the relation between η and δ is the same for both Θ and Θ.

Quasi-convex approximation and the Subharmonic Addition Theorem
In this section, we present a final ingredient for the duality-monotonicity method for potential theoretic comparison; namely, the so-called Subharmonic Addition Theorem. Roughly, it states that if one has a jet addition formula for subequations G, F and H in J 2 (X), then one has a subharmonic addition relation for the associated spaces of subharmonics. This implication will take on considerable importance when combined with the fundamental monotonicity-duality formula of jet addition noted in (2.19) Many results about F -subharmonic functions u, including the implication (4.1) ⇒ (4.2), are more easily proved if one assumes that u is also locally quasi-convex. Then, one can make use of quasi-convex approximation by way of sup-convolutions to extend the result to semicontinuous u. In general, when the subequations have variable coefficients, the quasi-convex approximation will be C 2 -perturbation (with small norm) of the sup-convolution. The quasi-convex approximation and subharmonic addition theorems in this section were essentially given in [21], and extend those known for for fiberegular subequations independent of the gradient [5,6].
4.1. Quasi-convex approximations. We begin by recalling some basic notions.
Such functions are twice differentiable for almost every 8 x ∈ X by a very easy generalization of Alexandroff's theorem for convex functions (the addition of a smooth function has no effect on differentiability). This is one of the many properties that quasi-convex functions inherit from convex functions. See [20] for an extensive treatment of quasi-convex functions. Quasi-convex functions are used to approximate u ∈ USC(X) (bounded from above) by way of the sup-convolution, which for each ε > 0 is defined by One has that u ε is 1 ε -quasi-convex and decreases pointwise to u as ε ց 0. Now, making use of the the uniform translation property of Theorem 3.3, we will prove the quasiconvex approximation result which is needed for the proof of the Subharmonic Addition Theorem in the case of fiberegular M-monotone subequations. This approximation result substitutes the constant coefficient result of [9, Theorem 8.2].
Remark 4.3. The approximating function u ε θ is quasi-convex, since it is the sum of a quasi-convex term, namely u ε , and a smooth term, namely θψ, with Hessian bounded from below.
Proof of Theorem 4.2. By the uniform translation property (Theorem 3.3), we know that By the sliding property (see Proposition B.4(iv)) we also have F ε := u z;θ − 1 2ε |z| 2 : |z| < δ ⊂ F (Ω δ ), and this family is locally bounded above. Therefore, by the families-locally-bounded-above property of (see Proposition B.4(vii)), the upper semicontinuous envelope v * ε of its upper envelope v ε := sup w∈Fε w belongs to F (Ω δ ). Now, a basic property of the sup-convolution is that it can also be represented as (for example, see [9, Section 8]): Hence, using the bound |u| M , by choosing one has and thus u * ε := (sup w∈Gε w) * = u ε since u ε is upper semicontinuous. The desired conclusion now follows by noting that v * ε = u * ε + θψ.

4.2.
Subharmonic addition for fiberegular M-monotone subequations. We will now make use of the quasi-convex approximation result of Theorem 4.2 to prove subharmonic addition. Given the local nature of the definition of subharmonicity, we are going to use the following local argument: in order to prove that F (X) + G(X) ⊂ H(X) it suffices to prove that F (B) + G(B) ⊂ H(B) for one small open ball B about each point of X. Therefore we will be in the situation where Ω = B ⊂ X can be chosen in such a way that M indeed admits a strict approximator on Ω: for every x ∈ X, it suffices to consider a quadratic strict M-subharmonic on B r (x), for some r > 0, (which we know there exists thanks to the topological property (T)) and then set B = B r/2 (x). In order to better understand the role of the assumptions of fiberegularity and M-monotonicity on the subequations, perhaps it is useful to review the argument in the constant coefficient case, as given in [7,Theorem 7.1]. Suppose that u ∈ F (X) and v ∈ G(X) for a pair of subequations F and G and suppose that there exists a third subequation H with F + G ⊆ H. As noted above, since the definition of H-subharmonic is local, in order to show that u + v ∈ H(X) it is enough to show that u + v ∈ H(U x ) for some open neighborhood U x of each x ∈ X. At this point, it is known [7, Remark 2.13] that if one chooses the U x 's to be small enough, then property (T) ensures the existence of smooth (actually, quadratic) subharmonics ϕ x ∈ F (U x ) and ψ x ∈ G(U x ). This is useful in order to apply another elementary property of the family of F -subharmonics (or G-subharmonics), namely the maximum property [7, Proposition D.1(B)] (see also Proposition B.4(ii)). This property says that: u, v ∈ F (X) ⇒ max{u, v} ∈ F (X). Applying the maximum property to the pairs (u, ϕ x −m), where ϕ x − m and ψ x − m are subharmonic by the negativity property, one obtains two approximating truncated sequences of subharmonics u m ∈ F (U x ), v m ∈ G(U x ), which are bounded on U x and decrease to the limits u, v, respectively as m → ∞. The boundedness on U x allows one to apply [9,Theorem 8.2] in order to produce, via the sup-convolution, two sequences of approximating quasi-convex subharmonics u ε m , v ε m , which are decreasing with pointwise limits u m , v m , respectively. Finally, one can now apply the Subharmonic Addition Theorem for quasiconvex functions [11, Theorem 5.1] and the decreasing sequence property [9, Section 4, property (5)] (or [7, Proposition D.1(E)], or Proposition B.4(v)) in order to conclude the proof.
The only obstruction to generalizing this constant coefficient proof to the case of variable coefficients is the need for a variable coefficient version of the constant coefficient quasi-convex approximation result of [9,Theorem 8.2]. All of the other steps are known to be valid also in the variable coefficient case: the local existence of smooth subharmonics easily follows (see [21,Remark 4.6] or [20, Remark 2.1.6] ) from the triad of topological conditions (T) which one requires a subequation to satisfy (cf. [10, Section 3]); the maximum property is straightforward and the decreasing sequence property can be proven essentially as in [9], by using the Definitional Comparison Lemma B.2 (see Proposition B.4). Therefore if one uses Theorem 4.2 instead of [9,Theorem 8.2], one has all the ingredients in order to carry out essentially the same proof.
Actually, it is worth noting one final thing: the parameters θ, ε, δ in (4.5), which are to be sent to 0, are linked in such a way that a priori, and in general one may suppose so, δ ց 0 as θ ց 0 (cf. (3.12) and the def. of δ), it is possible to let ε ց 0 with θ, δ fixed (cf. (4.7)), letting θ ց 0 would force ε ց 0 as well (cf. the relationships recalled above).
This suggests that one should first let ε ց 0 and then θ ց 0 (and thus δ ց 0). Also, we have no a priori information on the sign of the perturbing strict approximator in (4.5), namely θψ; hence one cannot use the decreasing sequence property in order to deal with the limit θ ց 0. Luckily enough, again thanks to the Definitional Comparison Lemma, another elementary property can be easily extended to variable coefficients, namely the uniform limits property [7, Section 4, property (5')] (see Proposition B.4(vi)); and the reader shall notice that, after computing the (decreasing) limit of u ε θ as ε ց 0, one gets u + θψ, which uniformly converges to u as θ ց 0.
The theorem that we are going to state has a gradient-free analogue [6, Theorem 5.2], which has been proven by applying the same procedure, where [ Proof. We have already outlined how a proof can be performed. For the sake of completeness, we give a brief sketch. Without loss of generality, suppose that u ∈ F (X) and v ∈ G(X) are bounded; indeed, if not, it suffices to proceed as follows: • for each x ∈ X, consider some ball B := B ρ (x) and two quadratic subharmonics ϕ ∈ F (B) and ψ ∈ G(B); • for all m ∈ N, define u m := max{u, ϕ − m} and v m := max{v, ψ − m}; • prove the theorem for u m and v m ; • apply the decreasing sequence property as m → ∞. Finally, as noted at the beginning of this subsection, after possibly choosing a smaller ball B, property (T) assures that M admits a (quadratic) strict approximator on B, so that we may assume without loss of generality that M admits a strict approximator ψ on X.
Thanks to these reductions, we are in the situation where all the hypotheses of Theorems 3.3 and 4.2 hold. Therefore we know that there exist two nets of quasi-convex functions where the parameter δ is chosen as δ := min{δ F , δ G }, where δ F and δ G are those coming from Theorem 3.3, associated to the subequations F and G, respectively. By the Subharmonic Addition Theorem for quasi-convex functions [11, Theorem 5.1], one has Therefore, since we know that by letting ε ց 0 the decreasing sequence property yields Letting θ ց 0, by the uniform limit property and the fact that X δ ր X as δ ց 0, This is equivalent to u + v ∈ H(X), which is the desired conclusion.

Potential theoretic comparison by the monotonicity-duality method
In this section, we present a flexible method for proving comparison (the comparison principle) in a fiberegular M-monotone nonlinear potential theory. The method works with sufficient monotonicity; that is, when the (constant coefficient) monotonicity cone subequation M admits a strict approximator ψ on a given domain Ω ⋐ R n , which we recall is a function ψ ∈ USC(Ω) ∩ C 2 (Ω) that is strictly M-subharmonic on Ω. Using monotonicity and duality, comparison is a consequence of the following constant coefficient Zero Maximum Principle (ZMP). We give two versions. The first is the "elliptic" version in Theorem 6.2 of [7] which uses a boundary condition on the entire boundary. The second is a "parabolic" version which uses a boundary condition on a proper subset of the boundary and generalizes Theorem 12.37 of [7].
Theorem 5.1 (ZMP for dual constant coefficient monotonicity cone subequations). Suppose that M is a constant coefficient monotonicity cone subequation that admits a strict approximator on a domain Ω ⋐ R n . Then the zero maximum principle holds for M on Ω; that is, for all z ∈ USC(Ω) ∩ M(Ω).
If, in addition, the strict approximator ψ satisfies for some ∂ − Ω ⊂ ∂Ω, then the zero maximum principle holds in the following form: for all z ∈ USC(Ω) ∩ M(Ω).
Proof. The first statement has been shown in [7,Theorem 6.2]. To get its version on the "reduced" boundary ∂ − Ω, one may argue as follows. Since int M has property (N) and since M is a cone, one has εψ − m is strictly M-subharmonic on Ω for each m > 0 and each ε > 0. The following is a general result for fiberegular M-monotone nonlinear potential theories.

Theorem 5.2 (A General Comparison Theorem).
Let Ω ⋐ R n be a bounded domain. Suppose that a subequation F ⊂ J 2 (Ω) is fiberegular and M-monotone on Ω for some monotonicity cone subequation M. If M admits a strict approximator on Ω, then comparison holds for F on Ω; that is, for all u ∈ USC(Ω), F -subharmonic on Ω, and w ∈ LSC(Ω), F -superharmonic on Ω.
Proof. As noted in (2.14), by duality, w is F -superharmonic on Ω if and only the function v := −w is F -subharmonic in Ω. Hence the the comparison principle (CP) is equivalent to for all u ∈ USC(Ω) ∩ F (Ω) and v ∈ USC(Ω) ∩ F (Ω). Obviously, (CP ′ ) is equivalent to the zero maximum principle (ZMP) for z := u + v, sums of F and F -subharmonics. By elementary properties of the Dirichlet dual [10,7,21] one knows that monotonicity and duality gives the jet addition formula
As discussed in the introduction, the utility of the General Comparison Theorem 5.2 is greatly facilitated by the detailed study of monotonicity cone subequations in [7]. For the convenience of the reader, we redroduce that discussion here. There is a three parameter fundamental family of monotonicity cone subequations (see Definition 5. Then the comparison principle (CP) holds on Ω.

Characterizations of dual cone subharmonics
In this section, we will present characterizations of the subharmonics M(X) determined by the dual of a monotonicity cone subequation M. Before presenting the characterizations, a few remarks are in order.
First, interest in such characterizations comes from the fact that the space of dual subharmonics M(X) on an open subset X ⊂ R n associated to a constant coefficient monotonicity cone subequation M ⊂ J 2 plays a key role in the monotonicity-duality method for proving comparison through the subharmonic addition theorem if F (and hence F ) is a fiberegular M-monotone subequation. This reduces comparison on a domain Ω ⋐ X to the zero maximum principle (ZMP) for M-subharmonics, which is in turn implied by the existence of a strict approximator ψ ∈ C 2 (Ω) ∩ C(Ω) (a strict M-subharmonic on Ω). Moroever, by (6.1), M(X) contains the differences of all F -subharmonics and F -superharmonics and M has constant coefficients, even if F does not. Second, since if one enlarges the monotonicity cone M, the chances of finding a strict approximator improve, while the space M(X) reduces, yielding a weaker (ZMP). This "monotonicity" in the family of monotonicity cones (6.2) will be used in the characterizations we present. Third, since In this case we will write u ∈ SA(X). With Ω ⋐ X fixed, we will also denote by SA(Ω) = {u ∈ USC(Ω) : (6.6) holds for each a ∈ A(Ω)}. (6.7) With respect to these definitions we will address two problems. where SA(Ω) is defined as in the second part of Definition 6.1.
Before presenting some motivating examples and the general results, a few remarks are in order.
Remark 6.2. A solution to Problem 1 will automatically solve Problem 2 for each Ω ⋐ X. We will see that a key role is played by domains Ω such that there exists a C 2 -strictly M-subharmonic function on Ω. (6.10) The property (6.10) holds for arbitrary Ω for many subequation cones M, but not all. Moreover, as noted at the beginning of the section, we are interested in the validity of the (ZMP) for M on Ω, so Problem 2 is interesting in its own right.
Remark 6.3. In both versions, there is an obvious "monotonicity property" since increasing the test functions a makes the sub-property (6.6) more restrictive. Hence the inclusion M(Ω) ⊂ SA(Ω) (6.12) is made easier for "smaller" classes A, while enlarging A will sharpen (6.12) and help in the reverse inclusion M(Ω) ⊃ SA(Ω) (6.13) We now begin to discuss some motivating examples. As noted in Examples 2.5 and 2.6, a characterization of the form (6.8) of Problem 1 is already known for two of the elementary monotonicity cone subequations in the fundamental family, which we recall in the following two examples. (6.14) SA(X) with A defined by (6.14) is the space of subaffine functions. This example appears in connection with every pure second order subequation F and every pure second order (degenerate) elliptic operator F . SA(X) with A defined by (6.15) is the space of subaffine-plus functions. This example appears in connection with every gradient-free subequation F and every gradient-free proper elliptic operator F .
The next example of an elementary monotonicity cone subequation in the fundamental family is, by iteslf, not particularly interesting. However, we record it anyway to make another point about intersections.  We will give general characterization results which also give sufficient conditions under which (6.18) holds. We begin with a lemma on the "reverse inclusion" of (6.13) which exploits part (b) of the Definitional Comparison Lemma B.2. Proof. For the claim (6.19), we assume that u ∈ SA(X) and we show that u ∈ M(X) by using part (b) of the Definitional Comparison Lemma with v = −a quadratic. It is enough to show that for every x 0 ∈ X, there exist arbitrary small balls B ρ (x 0 ) ⋐ X such that for each quadratic a such that −a is strictly M-subharmonic on B ρ (x 0 ). But we have (6.22) on every ball for all quadratic a such that −a is merely M-subharmonic on B ρ (x 0 ) (by the hypothesis that u ∈ SA(X) with A defined by (6.20). For the intersection property (6.21), for each Ω ⋐ X, consider where the last equality is merely the observation that for quadratic (C 2 ) functions a, which is equivalent to J 2 x (−a) ∈ M k for each x ∈ Ω for k = 1, 2. By the first part, we conclude that S(A 1 ∩ A 2 )(X) = SA(X) ⊂ M 1 ∩ M 2 (X).
Notice that Lemma 6.8 implies that for each Ω ⋐ X one has also the reverse inclusions SA(Ω) ⊂ M(Ω) and S(A 1 ∩ A 2 )(Ω) ⊂ M 1 ∩ M 2 (Ω) (6.23) Next we give a lemma on the "forward inclusion" (6.12) and the forward inclusion in the intersection property (6.18) on Ω ⋐ X which satisfy property (6.10). Lemma 6.9. Suppose that Ω admits a C 2 strict M-subharmonic for some monotonicity cone subequation M ⊂ J 2 . Then the following hold. Putting together Lemma 6.8 and Lemma 6.9, we have the following general result, whose proof is immediate. Theorem 6.10 (Characterizing dual cone subharmonics). Suppose that M ⊂ J 2 is a monotoncity cone subequation. Then the following hold. Remark 6.11. In Theorem 6.10, provided that M admits a C 2 strict M-subharmonic, we have characterizations M(Ω) = SA(Ω) with A(Ω) some class of quadratic functions easily determined by M; those quadratics a such that −a is M-subharmonic. However, it is not said that the characterization is optimal since it is possible that For example, by applying Theorem 6.10 to Example 6.4 with M = M(P) the theorem gives A 2 (Ω) as those quadratics a such that −a is M(P)-subharmonic; that is a a concave quadratic. On the other hand, we know that the characterization holds for A 1 (Ω) chosen as affine functions.
Obviously affine quadratics are also concave and are the "minimal" concave quadratics. In this pure second order case, one has the deep study of Harvey-Lawson [13] involving edge functions. Such improvements in the general case would be interesting.
We now complete the discussion by presenting the characterizations of M-subharmonics for all monotonicity cone subequations M that belong to the fundamental family of cones introduced in [7]. The family was recalled and briefly discussed beginning with the definition in (5.5)-(5.6): where with γ ∈ [0, +∞), D ⊂ R n a directional cone (a closed convex cone with vertex at the origin and non-empty interior), and R ∈ (0, +∞]. We recall that in the limiting case R = +∞ we interpret the last inequality in (6.34) as We recall also that the family is fundamental in the sense that for each monotonicity cone subequation  Clearly A R,min (Ω) A R (Ω) and the quadratics in A R,min (Ω) are "minimal" in the sense that they are the most "concave" quadratics in A R (Ω). Also notice that taking the limit R → +∞ of  in the last three cases of (6.45) with a directional cone D R n .

Admissibility constraints and the Correspondence Principle
In this section, we will discuss how the potential theoretic comparison principles in nonlinear potential theory (using monotonicity, duality and fiberegularity) developed in the previous sections can be transported to many fully nonlinear second order PDEs. The equations we treat will be defined by a variable coefficient operator F ∈ C(G) with domain G ⊂ J 2 (X) which may or may not be all of J 2 (X). Moreover, we will treat operators F with dependence on all jet variables J = (r, p, A) ∈ J 2 with sufficient monotonicity with respect to some constant coefficient monotonicity cone subequation M. Hence, the operators will be proper elliptic with an additional monotonicity in the gradient variables, a concept that we will call directionality. It is gradient dependence with directionality that distinguishes the present work with respect to the pure second order and gradient free situations treated in [5] and [6] respectively. 7.1. Viscosity solutions of PDEs with admissibility constraints. We begin by recalling the class of operators with the necessary monotonicity required for the comparison principle. When there is gradient dependence, the additional monotonicity of directionality will also be required (see Definition 7.5). is said to be proper elliptic if for each x ∈ X and each (r, p, A) ∈ G x one has F (x, r, p, A) F (x, r + s, p, A + P ) ∀ s 0 in R and ∀ P 0 in S(n). (7.1) The pair (F, G) will be called a proper elliptic 10 (operator-subequation) pair. 10 Such operators are often refered to as proper operators (starting from [8]). We prefer to maintain the term "elliptic" to emphasize the importance of the degenerate ellipticity (P-monotonicity in A) in the theory.
The minimal monotonicity (7.1) of the operator F parallels the minimal monotonicity properties (P) and (N) for subequations F . It is needed for coherence and eliminates obvious counterexamples for comparison. This is explained for subequations after Definition 2.3. A given operator F must often be restricted to a suitable background constraint domain G ⊂ J 2 (X) in order to have this minimal monotonicity (the constrained case). The historical example clarifying the need for imposing a constraint is the Monge-Ampère operator F (D 2 u) = det(D 2 u), (7.2) where one restricts the operator's domain to be the convexity subequation G = P := {A ∈ S(n) : A 0}.
Remark 7.2. The scope of the constrained case is perhaps best illustrated by the more general Gårding-Dirichlet operators as discussed in Section 11.6 of [7], of which the Monge-Ampère equation (7.2) represents the fundamental case. This class of operators are constructed in terms of hyperbolic polynomials in the sense Gårding (see Definition 8.6). The unconstrained case, in which F is proper elliptic on all of J 2 (X) is the case usually treated in the literature and is perhaps best illustrated by the so-called canonical operators associated to subequations with sufficient monotonicity, as discussed in Section 11.4 of [7].
We now recall the precise notion of subsolutions, supersolutions and solutions of a PDE The notions again make use of upper/lower test jets which we recall are defined by (a) a function u ∈ USC(X) is said to be a (G-admissible) viscosity subsolution of F (J 2 u) = 0 on X if for every x ∈ X one has J ∈ J 2,+ x u ⇒ J ∈ G x and F (x, J) 0; (7.6) (b) a function u ∈ LSC(Ω) is said to be a (G-admissible) viscosity supersolution of F (J 2 u) = 0 on X if for every x ∈ X one has A function u ∈ C(X) is an (G-admissible viscosity) solution of F (J 2 u) = 0 on X if both (a) and (b) hold.
In the unconstrained case where G = J 2 (X), the definitions are standard. In the constrained case where G J 2 (X), the definitions give a systematic way of doing of what is sometimes done in an ad-hoc way (see [16] for operators of Monge-Ampère type and [22] for prescribed curvature equations.) Note that (7.6) says that the subsolution u is also G-subharmonic and that (7.7) is equivalent to saying that F (x, J) 0 for the lower test jets which lie in the constraint G x .
If G is fiberwise constant, that is, for some E ⊂ R × R n × S(n), then G-admissible viscosity sub/supersolutions will be for simplicity referred as E-admissible viscosity sub/supersolutions.

The Correspondence Principle.
A crucial point in a nonlinear potential theoretic approach to study fully nonlinear PDEs is to establish the Correspondence Principle between a given proper elliptic operator-subequation pair (F, G) and a given subequation F . This correspondence consists of the two equivalences: for every u ∈ USC(X) u is F -subharmonic on X ⇔ u is a subsolution of F (J 2 u) = 0 on X (7.8) and u is F -superharmonic on X on X ⇔ u is a supersolution of F (J 2 u) = 0, (7.9) where the subsolutions/supersolutions are in the G-admissible viscosity sense of Defintion 7.3. By the definitions, the equivalence (7.8) is the same as the following equivalence: for each x ∈ X one has In addition, the equivalence (7.9) is the same as the following equivalence: for each x ∈ X one has Using duality (2.7) and J 2,+ x (−u) = −J 2,− x u one can see that that the equivalence (7.12) holds if and only if one has compatibility int F = {(x, J) ∈ G : F (x, J) > 0}, (7.13) which for subequations F defined by (7.11) is equivalent to These considerations can be summarized in the following result.
Theorem 7.4 (Correspondence Principle). Suppose that F ∈ C(G) is proper elliptic and F , defined by the correspondence relation (7.11), is a subequation. If compatibility (7.13) is satisfied, then the correspondence principle (7.8) and (7.9) holds. In particular, u ∈ C(X) is a G-admissible viscosity solution of F (J 2 u) = 0 in X if and only if u is F -harmonic in X.
It remains to determine structural conditions on a given proper elliptic operator F ∈ C(G) for which the hypotheses of the Correspondence Principle hold. There are the two requirements. First, one needs that the constraint set F defined by the correspondence relation (7.11) is, in fact, a subequation. The fiberwise monotoncity properties (P) and (N) for F follow easily from the M 0monotonicity of the proper elliptic pair (F, G). More delicate is the topological property (T) and this will require additional monotonicity and regularity assumptions on the pair (F, G). Also, in order to discuss the equation F (J 2 u) = 0 on X the following non-empty condition on the zero locus of F is needed Γ(x) := J ∈ G x : F (x, J) = 0 = ∅ for each x ∈ X. (7.15) This assumption also insures that F x = ∅ for each x ∈ X. Second, one needs the compatibility (7.13) (or equivalently (7.14) if F is a subequation). This condition is usually easy to check in practice, where some strict monotonicity of F near the zero locus of F suffices. We now address the question of sufficient conditions for having the first requirement of the Correspondence Principle for a given proper elliptic operator F ∈ C(G); that is, under what (additional) conditions on the pair (F, G) will the constraint set F defined by the correspondence relation (7.11) be a subequation? We will, in fact, do more. We will find conditions for which the constraint set F is a fiberegular M-monotone subequation for some monotonicty cone subequation M of the pair (F, G). This will make the Correspondence Principle useful for proving comparison. To that end, we must impose the appropriate (additional) monotonicity on the operator-subequation pair (F, G). Definition 7.5 (M-monotone operators). Let M ⊂ J 2 be a (constant coefficient) monotonicty cone subequation and let G ⊂ J 2 (X) be either G = J 2 (X) or G J 2 (X) an M-monotone subequation. An operator F ∈ C(G) is said to be M-monotone if (7.16) The pair (F, G) will be called an M-monotone (operator-subequation) pair.
Notice that all M-monotone operators are proper elliptic since any subequation cone M ⊂ J 2 cone contains the minimal monotonicity cone M 0 = N ×{0}×P; therefore, (7.16) implies (7.1). Also note that in the gradient free case, any proper elliptic operator is M-monotone for the monotonicity subequation cone M := N × R n × P. This is the case treated in [6].
Given an M-monotone operator F ∈ C(G), the fiber map Θ of the constraint set F defined by the compatibility relation (7.11) will be M-monotone in the following sense. Remark 7.7. Notice that if F is an M-monotone subequation on X, then the fiber map defined Θ(x) := F x for each x ∈ X will be an M-monotone map in the sense of Definition 7.6. However, this definition does not assume that Θ is the fiber map of an M-monotone subequation. Sufficient conditions which ensure that it is will be given in Theorem 7.11 below. Now, under a mild regularity condition on an M-monotone operator F ∈ C(G) (with G fiberegular in the constrained case), the fiber map of the constraint set F defined by the compatibility relation (7.11) will be continuous.
Theorem 7.8 (Continuous M-monotone maps). Let F ∈ C(G) be an M-monotone operator with either G = J 2 (X) or G J 2 (X) a fiberegular (M-monotone) subequation. Assume that the pair (F, G) satisfies the following regularity condition: for some fixed J 0 ∈ int M, given Ω ⋐ X and η > 0, there exists δ = δ(η, Ω) > 0 such that Then the M-monotone map Θ : X → K (J 2 ) defined by is continuous.
In the unconstrained case, where G = J 2 (X), the constant fiber map Φ ≡ J 2 is trivially continuous ((7.20) for Φ holds for every δ Φ > 0) and hence it suffices to choose δ Θ = δ and use the regularity condition (7.18).
Remark 7.9. In Theorem 7.8, the structural condition (7.18) on F is merely sufficient to ensure that an M-monotone map Θ given by (7.19) is continuous. The (locally uniform) continuity of Θ is equivalent to the statement that: for any fixed J 0 ∈ int M, given Ω ⋐ X, and η > 0, there exists δ = δ(η, Ω) > 0 such that ∀x, y ∈ Ω with |x − y| < δ one has F (x, J) 0 and J ∈ G x =⇒ F (y, J + ηJ 0 ) 0. (7.22) This condition is weaker, in general, than the structural condition (7.18) and hence useful to keep in mind for specific applications (see, for example, the proof of [6, Theorem 5.11] in a pure second order example). On the other hand, the structural condition (7.18) can be more easily compared to other structural conditions on F present in the literature.
Remark 7.10. Notice that Theorem 7.8 is really a result about continuous M-monotone maps.
In particular, we are not making use of the topological property (T) of G. In fact, one could state a version of the theorem where Φ is merely a continuous M-monotone map such that the F ∈ C(Φ(X)) is M-monotone in the sense that The conclusion is that Θ : X → K (X) defined by is continuous. An approach of focusing merely on a background fiber map Φ (and not a background subequation G) was followed in the pure second order and gradient free cases in [5] and [6].
Finally, making use of property (T) for a background subequation G and natural non-degeneracy conditions, we have the following result. Then the constraint set F defined by the correspondence relation (7.11); that is, is a fiberegular M-monotone subequation. Moreover, the fibers of F are non-empty if one assumes the non-empty condition (7.15). Each fiber F x in not all of J 2 in the constrained case and also in the unconstrained case if one assumes J ∈ J 2 : F (x, J) < 0 = ∅ for each x ∈ X. (7.25) Proof. As already noted, F defined by (7.24) will satisfy properties (P) and (N) with fiber map which is M-monotone and continuous (by Theorem 7.8). Hence it only remains to show that F satisfies property (T), which we recall is the triad The fiberwise property (T2), one can apply Proposition 4.7 of [7] which says that (T2) holds provided that the fibers F x are closed and M-monotone. This leaves properties (T1) and (T3). It is not hard to see that if F is closed, then properties (T2) plus (T3) imply (T1) (see Proposition A.2). Hence for a M-monotone pair (F, G), the constraint set F defined by (7.24) will be a subequation if F is closed and satisfies (T3). Moreover, since the inclusion (int F ) x ⊂ int (F x ) is automatic for each x ∈ X, (T3) reduces to the reverse inclusion, which holds provided that F is M-monotone and fiberegular in the sense of Defintion 3.1. This fact is proved in Proposition A.5. Finally, by Theorem 7.8, F will be fiberegular if G is fiberegular provided that F satisfies the regularity condition (7.18).

Comparison principles for proper elliptic PDEs with directionality
In this section, we present comparison principles for M-monotone operators by potential theoretic methods which combine monotonicity, duality and fiberegularity. A general comparison principle will be presented which gives sufficient structural conditions on the operator F which ensure that F satisfies the correspondence principle (Theorem 7.4) with respect to some subequation constraint set F . The comparison principle for the operator F will follow from the general comparison principle (Theorem 5.2) satisfied by the subequation F . Representative examples will be given for the constrained case in Examples 8.2, 8.4 and 8.7. As discussed in the introduction, we are primarily interested in examples will have gradient dependence in order to distinguish them from known examples the the pure second order and gradient-free cases that one finds in [5] and [6], respectively. The needed monotonicity in the gradient variables is called directionality, which together with proper ellipticity is incorporated into the notion of M-monotonicity for some monotonicity cone such as M(γ, D, R) where D R n is a directional cone.
Throughout the section M will be a constant coefficient monotonicity cone subequation and X an open subset of R n . 8.1. A general comparison principle for PDEs with sufficient monotonicity. We begin with the general result.
Then, for every bounded domain Ω ⋐ X for which M admits ψ ∈ C 2 (Ω) ∩ USC(Ω) which is strictly subharmonic on Ω, the comparison principle for the equation F (J 2 u) = 0 holds on Ω; that is, which arise in many areas of mathematical analysis (as we will see below), and are currently the object of intense research activity, see for example [1,17] and the references therein. Our first representative example will be treated using the first part (the elliptic version) of Theorem 8.1 which "sees" the entire boundary.
Example 8.2 (Optimal transport). The equation arises in the theory of optimal transport, and describes the optimal transport plan from a source density f to a target density g. Here, we assume that f, g ∈ C(Ω) are nonnegative, and we require that g satisfies a (strict) directionality property with respect to some directional cone D ⊂ R n (a closed convex cone with vertex at the origin and non-empty interior). More precisely, we assume that g(p + q) ≥ g(p), for each p, q ∈ D (8.8) and that there exists q ∈ int D and a modulus of continuity ω : (0, ∞) → (0, ∞) (satisfying ω(0 + ) = 0) such that g(p + ηq) ≥ g(p) + ω(η), for each p, q ∈ D and each η > 0. (8.9) In other words, g needs to be increasing on D in the directions of D and strictly increasing in some direction q ∈ int D ⊂ R n . Notice also that (8.9) implies that g ≡ 0 since g(q) g(0) + ω(1) > 0.
Note that, in view of Examples 2.5 and 2.7, u is a M(D, P)-admissible subsolution of (8.7) if and only if it is a subsolution of the PDE in the standard viscosity sense, it is convex on Ω, and it is nondecreasing in the D • -directions; that is, Our next representative example will be treated using the second part (the parabolic version) of Theorem 8.1 which "sees" only a reduced (parabolic) boundary.
Example 8.4 (Krylov's parabolic Monge-Ampère operator). In [18], the following nonlinear parabolic equation is considered This equation is important in the study of deformation of surfaces by Gauss-Kronecker curvature and in Aleksandrov-Bakel'man-type maximum principles for (linear) parabolic equations. We have in this case a comparison principle with respect to the usual parabolic boundary of Ω, for convex functions that are monotone nonincreasing in the t-direction.
In preparation for the result, we introduce some notation as well as the relevant monotonicity cone subequation. As above, (x, t) ∈ R n+1 = R n × R will be used as global coordinates on the domain. We will also denote by p = (p ′ , p n+1 ) ∈ R n × R in the first order part of the jet space. For matrices A ∈ S(n + 1), A n ∈ S(n) will denote the upper left n × n submatrix of A.
The relevant constant coefficient monotonicity cone subequation on R n+1 is M(D n , P n ) := {(r, p, A) ∈ R × R n+1 × S(n + 1) : p n+1 ≤ 0 and A n ≥ 0}; (8.11) that is, the (n + 1)-th entry of p is nonpositive and the n × n upper-left submatrix A n of A is nonnegative. Notice that M(D n , P n ) is clearly a convex cone with vertex at the origin and nonempty interior, which ensures the topological property (T1). Negativity (N) is trivial as M(D n , P n ) is independent of r ∈ R and positivity (P) also holds. Hence M(D n , P n ) is a (constant coefficient) monotoncity cone subequation.
be nonnegative with Ω ⋐ R n a bounded domain. Then, the parabolic comparison principle holds for M(D n , P n )-admissible sub/supersolutions u, v of (8.10).
Proof. As in the previous example, the idea is to apply Theorem 8.1. This time we will make use of the parabolic version applied to The assumptions (i), (ii), (iii), and (iv) are checked in the same fashion as done in the previous example.
Denoting by X = Ω × (0, T ), it remains to show that there exists ψ ∈ C 2 (X) ∩ USC(X) which is strictly M(D n , P n )-subharmonic on X and satisfies The function so we deduce the comparison principle in the form (CP -).
In view of Examples 2.5 and 2.7, u is a M(D n , P N )-admissible subsolution of (8.7) if and only if it is a subsolution of the equation in the standard viscosity sense, it is convex in the x variable, and it is nondecreasing in the {p n+1 ≤ 0} • -directions, which means that it is nonincreasing in the t variable.
Clearly, more general PDEs of the form could be addressed in a similar way (under suitable monotonicity assumptions), as well as "standard" parabolic equations ∂ t u = F (x, t, u, D x u, D 2 x u).

8.3.
Equations modelled on hyperbolic polynomials. Next we present perhaps the simplest meaningful example of a Gårding-Dirichlet operator, which are defined via hyperbolic polynomials in the sense of Gårding, as mentioned in Remark 7.2. We begin with the definition.
Definition 8.6. A homogeneous polynomial g of degree m on a finite dimensional real vector space V is called hyperbolic with respect to a direction a ∈ V if g(a) > 0 and if the one-variable polynomial t → g(ta + x) has exactly m real roots for each x ∈ V .
There are many examples of nonlinear PDEs that involve hyperbolic polynomials. The most basic example is the Monge-Ampère operator where g(A) = det A for A ∈ S(n) which is hyperbolic in the direction of the identity matrix I. A systematic study of the relationship between the Gårding theory of hyperbolic polynomials and pure second-order equations has been carried out in [12] (see also [7]). See also Section 11.6 of [7].
In the following example, we observe that the theory of hyperbolic polynomials is flexible enough to cover equations on the whole 2-jet space, providing a natural notion of monotonicity. As before, we focus our attention on the gradient dependence.
Example 8.7. On a bounded domain Ω ⊂ R 2 , we consider the equation x − u 2 y = 0, (8.12) which builds upon perhaps the simplest hyperbolic polynomial g(p 1 , p 2 ) = p 2 1 − p 2 2 . Since g is hyperbolic in the direction e = (1, 0) ∈ R 2 , a general construction of Gårding yields a monotonicity cone for the operator F (x, r, p, A) = g(p). In this example, the negatives of the two real roots t → g(t(1, 0) + p) can be ordered and are called the Gårding e-eigenvalues of g. Since g(e) = 1 (which can always be arranged by normalization since g(e) > 0) one has so that the first order differential operator defined by the degree two polynomial g (which is ehyperbolic) is the product of two Gårding e-eigenvalues of g. Gårding's theory also says that the closed Gårding cone must be convex and is characterized by the fact that its interior is the connected component of {g = 0} which contains e (both facts are clearly true here). Finally, since the closed convex cone with vertex at the origin Γ ⊂ R n has nonempty interior, , is a constant coefficient pure first order monotonicity cone subequation on R 2 . Moreover it is easy to check that F is M-monotone on F := R 2 × M. Finally, F is compatible with F by [12, Proposition 2.6] (which one can also check directly).
It is then rather easy to produce strict M g -subharmonics, and therefore to apply Theorem 8.1 to deduce a comparison result. We shall further specialize our setting to Ω = (a, b) × (c, d), where comparison on a reduced boundary is possible. To this end, take which goes to −∞ as x → a + , and satisfies Therefore, we have the following statement.
Then the comparison principle holds for M g -admissible sub/supersolutions u, v of (8.12), that is, As in the previous examples, comparison principles for u 2 x − u 2 y = f (x, y) could be deduced similarly. It is worth noting that equations of this form arise in the theory of zero-sum differential games. Though by no means general, we believe that this example well illustrates how the theory of hyperbolic polynomials and nonlinear potential theories may interact through a general notion of monotonicity to yield comparison principles for a large class of nonlinear PDEs. To further emphasize the flexibility of Gårding theory, we notice that one can easily deduce results for inhomogeneous operators with a product structure such as F (x, r, p, A) := g 1 (r)g 2 (p)g 3 (A) − f (x), where f ≥ 0 and g 1 , g 2 , g 3 are hyperbolic polinomials on R, R n , S(n) respectively. Indeed, each g i furnishes its own Gårding cone Γ i , and it is easy to check the monotonicity of F with respect to M := Γ 1 × Γ 2 × Γ 3 .

8.4.
Examples of equations where standard structural conditions fail. As a final consideration, we will present a class of proper elliptic operators with directionality for which our Theorem 8.1 applies to give the comparison principle, but for which the standard viscosity structural condition [8, condition (3.14)] on the operators fails to hold in general (see Proposition 8.10 below). Simpler examples of variable coefficient pure second order operators have been discussed in [5,Remark 5.10].
The aforementioned condition (rewritten for F (x, r, p, A) which is increasing in A, according to our convention) is with f ∈ U C(Ω; [0, +∞)) and with M ∈ U C(Ω × R n ; S(n)) of the form with P ∈ U C(Ω; P) and b ∈ U C(Ω; R n ) such that there exists a unit vector ν ∈ R n such that b(x), ν 0 for each x ∈ Ω. (8.16) Notice that the required uniform continuity for f and M holds if they are continuous on an open set X for which Ω ⋐ X. One associates to F the candidate subequation with fibers which are clearly (R × {0} × P)-monotone and hence one has properties (N) and (P) for F . In order to conclude that F is indeed a subequation, by Theorem A.7, it suffices to show that F is fiberegular and M-monotone for some (constant coefficient) monotonicity cone subequation. To construct M define the half-spaces and define the cone This D is a directional cone for F . Indeed M (·, p + q) M (·, p) for all q ∈ D. Hence F is M(D, P)-monotone.
The comparison principle for F -admissible sub/supersolutions of the equation F = 0 then follows from Theorem 8.1 since every bounded domain Ω admits a quadratic striclty M(D, P)-subharmonic function ψ (as recalled in the proof of Proposition 8.3) and F satisfies the required conditions of the theorem (which we leave to the reader).
We now arrive to the main point of this subsection. While the comparison principle holds for the equation F = 0 of Example 8.9, if admissible "perturbation coefficients" b(x) and P (x) are chosen suitably, the standard viscosity condition (8.13) fails to hold. For simplicity we give an example in dimension two, but generalizations are clearly possible. Proposition 8.10. For some x 0 ∈ Ω ⋐ R 2 , suppose that the perturbation coefficients b ∈ U C(Ω; R 2 ) and P ∈ U C(Ω; P) satisfy: b has an isolated zero of order β ∈ (0, 3) at x 0 ; (8.17) , (8.18) where g ∈ U C(Ω) is nonnegative and satisfies g has an isolated zero of order γ ∈ β+1 2 , 2 at x 0 .
Then the condition (8.13) fails to hold.
Before giving the proof, we should note that the function h in (8.18) is not actually defined in x = x 0 , but since g 2 has an isolated zero of order 2γ > β + 1 > 1, h extends continuously by defining h(x 0 ) = 0.
Proof. The idea is to exploit the order of the isolated zeros of b and g in x 0 to take a suitable sequence {y n } ⊂ Ω converging to x 0 , along which one can find sequences of matrices {A n } and {B n } satisfying (8.14) which contradict the validity of the inequality(8.13).
Consider any sequence {y n } n∈N ⊂ Ω such that y n → x 0 with b(y n ) = 0 ∈ R 2 (b has an isolated zero in x 0 ) and such that b(y n ), x 0 − y n > 0 ∀n ∈ N.
Such a choice is possible thanks to condition (8.16). The desired matrices are defined by 2A n = B n := 0 0 0 g(y n ) −1 , n ∈ N.
Each pair A n , B n satisfies (8.14) with α = α n := (3g(y n )) −1 , as one easily verifies. By contradiction, assume that (8.13) holds. Along the sequence (y n , A n , B n ) one would have, as y n → x 0 , thus leading to a contradiction.
Appendix A. Monotonicity, fiberegularity and topological stability Given a fiberegular subequation F ⊂ J 2 (X) = X × J 2 on an open set X in R n which is Mmonotone for some constant coefficient monotonicity cone subequation M, one knows that the fiber map Θ : (X, | · |) → (K (J 2 ), d H ) defined by Θ(x) := F x , x ∈ X is continuous (taking values in the closed subsets K (J 2 )) and M-monotone in the sense that We address here the converse; that is, given a continuous M-monotone map is it true that The question is important in light of the Correspondence Principle of Theorem 7.4; that is, given a proper elliptic pair (F, G) one wants to know whether F having fibers implies that F is a subequation (and hence a well developed potential theory). If Θ is a continuous M-monotone map, then by definition the fibers of F x := Θ(x) are closed and one has M-monotonicity which implies properties (P) and (N) since M 0 := N × {0} × P ⊂ M. This leaves the topological property (T), which we recall is the triad In the case of constant coefficient subequations, the triad reduces to (T1); that is, to F being a regular closed set, and it is known [9,7] that such condition is equivalent to the reflexivity of the Dirichlet dual This is essentially a consequence of the fact that, for any open set O, the set S = O is regular closed. When F has variable coefficients, as noted in [10], the two conditions (T2)-(T3) involving the fibers are useful in order to be allowed to compute the dual fiberwise; that is, in order to have the following equality: Therefore it is easy to see that the full topological condition (T) (that is, conditions (T1)-(T3) together) yields the reflexivity of Dirichlet duals of variable coefficient subequations as well. Some facts concerning the topological conditions are in order.
Proposition A.1. Let F ⊂ X × J 2 ; then (T3) holds if and only if Proof. Equality(T3 ′ ) straightforwardly implies condition (T3). For the converse implication, one first notes that the inclusion always holds, so that we have that Hence, combining (A.5) and (A.6) yields By (A.7), one has the inclusion ⊃ in (T3 ′ ), while the opposite one is trivial by (A.4).
Proposition A.2. Let F ⊂ X × J 2 be closed. Then Proof. On always has the inclusion int F ⊂ F . On the other hand, assuming (T2) and (T3), where we also used Proposition A.1 for the last equality.
This shows that an equivalent formulation of property (T) would be to ask that F is closed and that (T2) and (T3) hold.
As for (T2) and (T3), it is easy to see that, in general, closed M 0 -monotone subsets of X × J 2 do not satisfy it. However, if one has more monotonicity, then that could be enough in order to guarantee For instance, the following holds.
Proposition A.3. Suppose that F ⊂ X × J 2 has closed fibers and suppose that there exists a subset M ⊂ X × J 2 where M satisfies property (T2) and Then F satisfies property (T2).
Proof. One has hence all the inclusions (and in particular the last one) are in fact equalities.
Remark A.4. A situation in which the hypotheses of Proposition A.3 are satisfied is that of F being M-monotone for some regular closed subset M ⊂ J 2 such that 0 ∈ M. For example, this holds if F is M monotone for a constant coefficient monotonicity cone subequation.
Condition (T3) requires a little more attention because it is the only one that relates the interior with respect to X × J 2 and the interior with respect to J 2 . To stress this fact, let us write This condition seems to be related to some sort of continuity of the fiber F x with respect to the point x. Here we prove that if F ⊂ J 2 (X) has fibers determined by a continuous M-monotone map Θ, then F satisfies (T3). Fix x ∈ X and J x ∈ int Θ(x) and let ρ > 0 such that B 2ρ (J x ) ⊂ Θ(x), where B denotes the ball in J 2 with respect to the norm ||| · |||. Let J ′ x ∈ B ρ (J x ), so that for some J 0 ∈ int M fixed. By Proposition 3.2(c), there exists δ > 0 such that This proves that It follows that (x, J x ) ∈ B δ (x) × B ρ (J x ) ⊂ int |y−x|<δ Θ(y), and since J x ∈ int Θ(x) is arbitrary, this proves that and thus, since x ∈ X is arbitrary, (A.8) follows, as desired.
Finally, the continuity of Θ also implies that F is closed.
Proposition A.6. Suppose that there exists Θ as above (not necessarily M-monotone). Then F is closed (in X × J 2 ).
Proof. Let (x, J) ∈ F . Then x ∈ X and there exist sequences x k → x and J k → J such that J k ∈ Θ(x k ) for all k ∈ N. Since by continuity Θ(x) = lim x k →x Θ(x), where the limit is computed with respect to the Hausdorff distance d H , it is known (cf. [2, Exercise 7.4.3.1]) that Θ(x) = {J ′ ∈ J 2 : ∃{J ′ k } k∈N such that J ′ k ∈ Θ(x k ) ∀k ∈ N and J ′ k → J ′ }. Hence J ∈ Θ(x), yielding (x, J) ∈ F .
We now can affirm that the answer to the question (A.3) is yes. Then F is an M-monotone subequation on X.
Proof. By definition, F is M-monotone; that is, F + M ⊂ F . Also, F has nontrivial, and closed, fibers. Therefore the proof now amounts to showing that F satisfies the triad of topological properties. By Proposition A.3 and Remark A.4, F satisfies (T2), by Proposition A.5, F satisfies (T3); this means that F satisfies (T2) and (T3). By Proposition A.6, F is closed, and thus, by Proposition A.2, F satisfies(T1) as well.
[9], whose proofs are somewhat reformulated in [20] making more explicit use of the definitional comparison of Lemma B.2.
We begin with the first tool which is very useful when one seeks to check the validity of subharmonicity at a point by a contradiction argument. More precisely, if u fails to be subharmonic at a given point, then one must have the existence of a bad test jet at that point, as stated in the following lemma. This criterion is essentially the contrapositive of the definition of viscosity subsolution, when one takes strict upper contact quadratic functions as upper test functions (see [7, Lemma 2.8 and Lemma C.1]).
Lemma B.1 (Bad Test Jet Lemma). Given u ∈ USC(X), x ∈ X and F x = ∅, suppose u is not F -subharmonic at x. Then there exists ε > 0 and a 2-jet J / ∈ F x such that the (unique) quadratic function ϕ J with J 2 x ϕ J = J is an upper test function for u at x in the following ε-strict sense: u(y) − ϕ J (y) −ε|y − x| 2 ∀y near x (with equality at x). (B.1) The second tool is a comparison principle whose validity characterizes the F -subharmonic functions for a given subequation F . It states that comparison holds if the function z in (ZMP) is the sum of a F -subharmonic and a C 2 -smooth and strictly F -subharmonic. It is called definitional comparison because it relies only upon the "good" definitions the theory gives for F -subharmonics and for subequations F (which include the negativity condition (N) that is important in the proof). It was stated and proven in a context of constant coefficient subequations in [7, Lemma 3.14].
Lemma B.2 (Definitional Comparison). Let F be a subequation and u ∈ USC(X). That is to say, in order to show by (b) that u is subharmonic on X one proves that, for each x ∈ X there is a neighborhood Ω ⋐ X of x where (u + v)(x 0 ) > 0 for some x 0 ∈ Ω =⇒ (u + v)(y 0 ) > 0 for some y 0 ∈ ∂Ω (B.4) for every v ∈ USC(Ω) ∩ C 2 (Ω) which is strictly F -subharmonic on Ω. Conversely, one can also infer that the implication (B.4) holds whenever one knows that u is subharmonic on X. In situations where we are interested in proving the subharmonicity of a function which is somehow related to a given subharmonic, this helps to close the circle (for example, see the proofs of Theorem 3.3 or Proposition B.4).
The last tool is a collection of elementary properties shared by functions in F (X), the set of F -subharmonics on X. They are to be found in [9,Section 4] for pure second-order subequations, in [10, Theorem 2.6] for subequations on Riemannian manifolds, in [7, Proposition D.1] for constantcoefficient subequations. By invoking the Definitional Comparison Lemma B.2 one can perform most of the proofs along the lines of those of Harvey-Lawson [9]. This is done in [20]. More precisely, one uses the definitional comparison in order to make up for the lack, for arbitrary subequations, of a result like [9,Lemma 4.6].
Appendix C. Some facts about the Hausdorff distance We briefly recall a few facts about the Hausdorff distance which we have used in the discussion of fiberegularity in Subsection 3. The reader can consult [2] for further information. It is immediate to see that the upper semicontinuous envelope operator * : g → g * is the identity on the set of all upper semicontinuous functions. Also, we called Perron function the upper envelope of the family F , since F is a family of subharmonics. In addition, if we consider (M, d) = (J 2 , ||| · |||), we know that the Hausdorff distance is infinite in another case as well.
Proof. It suffices to show that E c contains balls of arbitrarily large radius, so that no finite enlargement of E can exhaust J 2 . Note that by the definition of the Dirichlet dual, At this point, it suffices to show that int M contains balls of arbitrarily large radius. To see this, fix J 0 ∈ int M and without loss of generality suppose that B 1 (J 0 ) ⊂ M; 11 note that one has tJ 0 ∈ int M for any t > 0 and B t (tJ 0 ) ⊂ int M ∀t > 0.