On the Characteristic Curvature Operator

We introduce the Characteristic Curvature as the curvature of the trajectories of the hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces and by using the Second Fundamental Form we relate it to the Classical and Levi Mean Curvature. Then we prove existence and uniqueness of viscosity solutions for the related Dirichlet problem and we show the Lipschitz regularity of the solutions under suitable hypotheses. Moreover we prove a non existence result on the balls when the prescribed curvature is a positive constant. At the end we show that neither Strong Comparison Principle nor Hopf Lemma do hold for the Characteristic Curvature Operator.


Introduction
In this paper we introduce the Characteristic Curvature as the curvature of the trajectories of the Hamiltonian vector field with respect to the normal direction to the isoenergetic surfaces. Namely, let H be a smooth Hamiltonian function on R n+1 × R n+1 equipped with its standard symplectic structure J; then the level set M of H, corresponding to some noncritical energy value E, is a smooth hypersurface in R 2n+2 . The Hamiltonian vector field X H is the tangent vector field to M , defined by X H := JDH. The orbits of X H are the critical points of the Action functional defined on a suitable space of curves; therefore they represent the trajectories of the motion in the generalized phase space. In particular they are curves on M : we will define the characteristic curvature C M as the normalized curvature of these curves with respect to the unit normal direction to M ; we will say that M is strictly C-convex if C M > 0. Later, since M is a real hypersurface in C n+1 , by using the Second Fundamental Form and the Levi Form we relate C M to the Classical Mean Curvature H M and to the Levi Mean Curvature L M . We want explicitly to note that the characteristic curvature C M can be used to obtain characterization properties: in fact, following some results obtained by Hounie and Lanconelli ([7], [8]) in which they prove Alexandrov type theorems for Reinhardt domains by using the Levi Mean Curvature, it is proved in [11] an analogous symmetry result for Reinhardt domains, starting from the characteristic curvature C M . Denoting by T the related differential operator, we will find the explicit expression T u = tra A(Du) D 2 u , being A a symmetric matrix defined in the sequel. The characteristic curvature operator T is a quasilinear (highly) degenerate elliptic operator on R 2n+1 : in fact the principal part has 2n distinct eigenvectors corresponding to the eigenvalue zero and only one direction of positivity. Let Ω be a bounded open set in R 2n+1 , then under suitable hypotheses we will prove existence and uniqueness of viscosity solutions for the associated Dirichlet Problem, with prescribed curvature function k ∈ C(Ω × R): where ϕ ∈ C(∂Ω). In order to do that we will use the classical tools introduced by Crandall, Ishii, Lions in [3], [9] and we will give geometric sufficient conditions on Ω and on the prescribed curvature k in order to ensure the existence of sub-and supersolutions for (DP ). Namely, if we denote by Ω c := ∂Ω × R, the cylinder type hypersurface in R 2n+2 , then we will assume: let Ω c be a strictly C-convex hypersurface, then sup s∈R k(x, s) < C Ωc x , for every x ∈ ∂Ω; (1) and let R be the radius of the smallest ball containing Ω, then We will prove the following result: (1) and (2) hold. If k is either strictly increasing with respect to u or non-decreasing with respect to u but independent of x, then there exists a unique viscosity solution for (DP ).
Later we will prove the Lipschitz regularity of solutions: first we use a Bernstein method to obtain a gradient bound for the solutions of the regularized equation and then we use a limit process argument. In particular we need a slightly stronger assumption than (1): let Ω c be a strictly C-convex hypersurface such that there exists a defining function for Ω, ρ ∈ C 2,α , 0 < α < 1, with ρ > 0 on ∂Ω, then Remark 1.2. The hypothesis of having a defining function with ρ > 0 is obviously fulfilled if ∂Ω is strictly convex; it is also satisfied if the cylinder Ω c is strictly pseudoconvex as hypersurface in C n+1 .
Therefore we prove: Let us suppose that the hypotheses (2) and (3) hold. Let then (DP ) has a Lipschitz continuous viscosity solution. Moreover, if k is either strictly increasing with respect to u or non-decreasing with respect to u but independent of x then the solution is unique.
We then show a non-existence result on balls when the prescribed curvature is a positive constant. Similar results were proved by Slodkowski and Tomassini in [13] for the Levi equation in the case n = 1; by Martino and Montanari in [12] for the Mean Levi Curvature; by Slodkowski and Tomassini in [14] and by Da Lio and Montanari in [4] for the Levi Monge Ampère equation. At the end, by mean of two counterexamples, we will show that neither the Strong Comparison Principle nor the Hopf Lemma hold for the operator T . This is substantial difference between the highly degenerate Characteristic operator and the Levi Curvature operators, for which Lanconelli and Montanari in [10] proved the Strong Comparison Principle: indeed the principal part of Levi Curvature operators is degenerate only with respect to one direction and when computed on strictly pseudoconvex functions, the 2n vector fields, respect to which the operator is strictly elliptic, satisfy the Hörmander rank condition.
Acknowledgement I wish to thank Professor Annamaria Montanari and Professor YanYan Li for their useful suggestions and comments on the preparation of this work. Moreover this paper was completed during the year that I spent at the Mathematics Department of Rutgers University: I want to express my gratitude for the hospitality and I'm grateful to the Nonlinear Analysis Center for its support.

The characteristic curvature
Here we recall some known facts and we refer for instance to [6] for a full detailed exposition. Let us consider z = (x, y) ∈ R n+1 × R n+1 . We will denote by λ = (1/2) n+1 k=1 (y k dx k −x k dy k ) the standard Liouville differential 1-form and by ω := dλ the canonical symplectic 2-form. We will also denote by g the standard scalar product in R 2n+2 , and by J the canonical symplectic matrix in R 2n+2 . Let us consider a smooth Hamiltonian function H : R n+1 × R n+1 → R, z = (x, y) −→ H(x, y) = H(z), and let M be the isoenergetic hypersurface in R 2n+2 defined by M = {z ∈ R 2n+2 : H(z) = E}, with DH(z) = 0 for all z ∈ M , where E is some constant. The trajectories of motion are solutions of the following first order system (Hamilton) Moreover if γ solves (6), then γ ⊆ M . Now we introduce the Hamiltonian vector field X H z := JDH(z); then the Hamilton system (6) rewrites aṡ γ = X H γ . We explicitly remark that the direction given by the Hamiltonian vector field only depends on M and J. By taking the restriction of ω on T M , one has rank(ω| T M ) = 2n and dim(ker(ω| T M )) = 1 We introduce then the following one-dimensional subspace of the tangent space: Since therefore X H z ∈ K z , ∀z ∈ M , and its orbits are characteristic curves on M .
Remark 2.1. Since ker(ω| T M ) is a one-dimensional subspace and X H (which never vanishes) always belongs to ker(ω| T M ), then a smooth curve γ ⊆ M is characteristic, up to reparametrization, if and only ifγ = X H .
We want to compute the curvature of the characteristic curves with respect to the normal direction to M .
Definition 2.2. Let ε > 0 and let γ : (−ε, ε) → M be any smooth curve such that γ(0) = z ∈ M andγ(0) ∈ K γ(0) . We will call the characteristic curvature of M at z the following We can obtain a formula for the characteristic curvature explicitly involving only the characteristic curves. In fact, let γ ⊆ M be a characteristic curve, then a unit normal direction along γ is given by N γ = Jγ/|γ|, and therefore Remark 2.3. By the previous formula we can see that the characteristic curvature is a scalar invariant under (rigid) symplectic diffeomorphisms.
We will add some explicit examples at the end of the paper.

Relation with the Classical and Levi Mean Curvature
Let M be a smooth real hypersurface in C n+1 and let us identify We can compare the Levi Form with the Second Fundamental Form (see [2], Chap.10, Theorem 2): be an orthonormal basis of the horizontal space HM ; then the Second Fundamental Form has the following structure Therefore a direct computation leads to the relation between H M , L M and C M :

The operator
Here we find an explicit formula for C M that involves a defining function f . A direct computation shows that for any z ∈ M we have: where A is the following (2n + 2) × (2n + 2) symmetric matrix: We define the characteristic curvature operator T as the differential second order operator acting on f in the following way: We are interested in finding an expression for T when we locally consider the hypersurface M as the graph of some function u : R 2n+1 ⊇ Ω → R such that (ξ, u(ξ)) ∈ M for all ξ ∈ Ω. In order to do that, we set x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ), x n+1 = t, y n+1 = s, ξ = (x, y, t) and we take as defining function Then we have T u := 1 where A is the following symmetric matrix: The characteristic curvature operator T is a second order quasilinear (highly) degenerate elliptic operator on R 2n+1 : in fact, by (11) we see that the following 2n independent vector fields are eigenvectors for A with eigenvalue identically equals to zero; instead the vector field −u y 1 ∂ x 1 − u yn ∂ xn + u x 1 ∂ y 1 + u xn ∂ yn + ∂ t is an eigenvector for A with eigenvalue equals to (1 + |u x | 2 + |u y | 2 ). For the sake of simplicity we will call A(p) =

Viscosity solutions
Here we prove Theorem (1.1). We refer the reader to [3], [9] for a complete exposition regarding the theory of viscosity solutions. If k is a prescribed continuous function, non-negative and strictly increasing with respect to u, then F is proper according the definition in [3] and then the comparison principle for F holds. Anyway, since we are interested even at the case when the characteristic curvature is constant, we would like to have a comparison principle for F also when k is not strictly increasing with respect to u, but it does not depend on x. We will adapt the proof for the strictly monotone case: in order to do that we need two standard lemmas and we refer the reader to [3] for the proofs.
Lemma 5.2. Let Σ i ⊆ R n i be a locally compact set and u i ∈ U SC(Σ i ), for i = 1, . . . , k. We define: . . . , x k ) ∈ Σ, and n 1 + . . . + n k = 2n + 1. Let us suppose that x = ( x 1 , . . . , x k ) is a local maximum for w(x) − ϕ(x), where ϕ ∈ C 2 in a neighborhood of x. Then, for every ε > 0, there exist Λ i ∈ S(n i ) such that with Φ = D 2 ϕ( x) ∈ S(2n + 1) and the norm for Φ is: We can prove then the following result: where g ∈ C 2 and g , g > 0. We have (tra(A(p)) = g > 0 Moreover we choose g in such a way that h ∞ < +∞. Our aim is to show that sup We suppose by contradiction that for all m large enough we have Since we have u(y) ≤ u(y) for all y ∈ ∂Ω, such a maximum is achieved at an interior pointx (depending on m). For all ε > 0 let us consider the auxiliary function w ε (x, y) = u m (x)−v(y)− |x−y| 2 2ε . Let (x ε , y ε ) be a maximum of w ε in Ω×Ω. By Lemma (5.1) we get as ε → 0, up to subsequences, x ε , y ε →x ∈ Ω, . We can suppose without restriction thatx = 0. Sincex is necessarily in Ω, for ε small enough we have x ε , y ε ∈ Ω. By Lemma (5.2), there exist X, Y ∈ S(n) such that, if p ε := (xε−yε) ε , we have Moreover u m is a strictly viscosity subsolution of ) Then by using (12) we have Now we note that f (p m ε ) ≈ f (p ε ) as m → ∞, and by hypotheses on k and Lemma (5.1) we get k(u( x) − k(u( x)) ≤ 0, as ε → 0. Therefore by choosing m = ε −2 and taking the limit as ε → 0 we obtain a contradiction.
Proof. (of Theorem 1.1) Since we have comparison principle for both cases, by the Perron type theorem in [9] (Proposition II.1), we have that if there exist a subsolution u and a supersolution u for (DP ) such that u = u = ϕ on ∂Ω, then there exists a unique viscosity solution for (DP ). Therefore we are interested in finding explicit sub-and supersolutions for (DP ). Let ρ ∈ C 2 be a defining function for Ω. Let V 0 = {x ∈ R 2n+1 : −γ 0 < ρ(x) < 0}, γ 0 > 0 such that for every 0 ≤ γ ≤ γ 0 the cylinder Ω γ c still satisfies (1), where Ω γ = {x ∈ R 2n+1 : ρ(x) < −γ}. Let {ϕ ε } ε>0 be a sequence of smooth functions uniformly convergent to ϕ on ∂Ω; let finally ϕ ε be a smooth extension of ϕ ε on Ω. We define u ε (x) = ϕ ε (x) + λρ(x) and u ε (x) = ϕ ε (x) − λρ(x), with λ > 0. It holds u ε = u ε = ϕ ε on ∂Ω and for λ large enough we have u ε ≤ u ε on V 0 . Now by (1), for every x ∈ V 0 , one has: Let x 0 be the center of the smallest ball B(x 0 , R) containing Ω and let us introduce the function h( Therefore v ε and v ε are respectively sub-and supersolution of (DP ) with boundary datum ϕ ε . Then there exists a unique viscosity solution of (DP ). From comparison principle Since viscosity solutions are stable with respect to uniform convergence (see [3]) then u ε uniformly converges to the unique solution of (DP).

Lipschitz viscosity solutions
In this section we are looking for Lipschitz continuous viscosity solutions of (DP ). We will regularize in elliptic way our operator in order to obtain a smooth solution u ε ; then we will prove a uniform gradient estimate for Du ε by using a Bernstein method and finally we will get our solution by taking the uniform limit of u ε . For 0 < ε ≤ 1, let us set A ε (p) := A(p) + εId; then A ε is strictly positive definite and is elliptic. We consider the following perturbed Dirichlet Problem: We prove: Proposition 6.1. Let k ∈ C 1 (Ω × R) and ϕ ∈ C 2,α (∂Ω), 0 < α < 1. If (4) and (5) hold, then (DP ε ) admits a solution u ε ∈ C 2,α (Ω) such that Proof. The first statement is a consequence of the ellipticity of F ε (see [5]).
By differentiating (14) with respect to x k , we get: We multiply by ∂ k u ε and we take the sum over k: By substituting in (15), we have By Schwarz theorem and by (14), it holds: Therefore by using (16) and hypothesis (5) we get We can apply the classic maximum principle for elliptic operators (see [5]) and we obtain that max Ω |v ε | = max ∂Ω |v ε |; therefore (13) holds.
Next we prove a non-existence result on balls, when the prescribed curvature is a positive constant, following the idea in [1].
Proposition 6.4. Let B ⊆ R 2n+1 be the ball with center x 0 and radius R and let us suppose that k is a positive constant. If u is a Lipschitz continuous viscosity solution of F = 0 in B, then necessarily it holds R ≤ 1/k.
Proof. Let 0 ≤ r ≤ R and let us consider the function φ(x) = M − r 2 − |x − x 0 | 2 , for some constant M . We have that φ ∈ C 2 (B) and By the Lipschitz regularity of u on B we can choose M such that u − φ has a maximum at an interior pointx ∈ B; then we get (u is a viscosity subsolution of F = 0 as well) F (x, u(x), Dφ(x), D 2 φ(x)) ≤ 0, that is: k ≤ tra( A(Dφ(x))D 2 φ(x)) = 1 r for every 0 ≤ r ≤ R. This ends the proof.

Some examples and counterexamples
Here we show by easy counterexamples that the Strong Comparison Principle and the Hopf Lemma do not hold for the characteristic operator T . We have in Ω \ {p}, p ∈ ∂Ω u(p) = v(p) p ∈ ∂Ω and ∂u ∂ν (p) = ∂v ∂ν (p) = 0.
Next we give some explicit examples of domains with the related characteristic curvature.  Example 7.5 (characteristic curvature of cylinder type domains -2). Let us consider H(x, y) = (1/2)(x 2 1 + y 2 1 ) as Hamiltonian function in R 2 × R 2 ; for any positive constant E the isoenergetic surface of H is a cylinder C 2 = S 1 R × R 2 with circles of radius R = √ 2E and we have C C 2 = 1/R. Remark 7.6. By the previous two examples we see that the two isometric hypersurfaces C 1 and C 2 in R 2 × R 2 have different characteristic curvature: indeed the isometry that exchanges x 2 to y 1 is not a symplectic diffeomorphism.