Nonlocal Lagrange multipliers and transport densities

We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs b\,u)+\bs d\cdot D^su+c\,u , \end{equation*} with integrable data, in the space $\Lambda^{s,p}_0(\Omega)$, which is the completion of the set of smooth functions with compact support in a bounded domain $\Omega$ for the $L^p$-norm of the distributional Riesz fractional gradient $D^s$ in $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. For this purpose, we also develop some relevant properties of the spaces $\Lambda^{s,p}_0(\Omega)$, including the limit case $p=\infty$ and the continuous embeddings $\Lambda^{s,q}_0(\Omega)\subset \Lambda^{s,p}_0(\Omega)$, for $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint.


Introduction
In a bounded open set Ω of R d , consider the model problem for the pair of functions (u, λ), where δ ≥ 0 is a constant, D denotes the gradient, D• denotes the divergence and f = f (x) is a given function.
For δ > 0, the problem (1.1)-(1.2),being equivalent to minimise the functional in the convex subset of H 1 0 (Ω) subjected to the constraint |Du| ≤ 1 in Ω, is well-known to model the elastoplastic torsion of a cylindric bar of cross section Ω, where λ is the respective Lagrange multiplier.In 1972, Brézis [10] has shown that, if f = const > 0 and Ω is simply connected, λ ∈ L ∞ (Ω) is unique and even continuous if Ω is convex.This was partially extended to more general strictly convex functionals than (1.3), by Chiadò Piat and Percivale [12], for f ∈ L p (Ω), p > d, obtaining a solution u in C 1,α (Ω), α = 1 − d p and λ as a positive Radon measure (see the survey [25], for references and more results).
In the degenerate case δ = 0, (1.1)-(1.2) are usually called the Monge-Kantorovich equations, as they appear in a classical mass transfer problem [17], where u and λ represent the transport potential and density, respectively.This is the dual problem of (1.3) with δ = 0 over all Lipschitz continuous functions with |Du| ≤ 1 and vanishing on ∂Ω.This same problem also arises in shape optimization [7], in the equilibrium configurations [6] and in the time discretisation of the growing sandpile problem [16].
In general, and specially in the case δ = 0 with more general gradients thresholds, the main difficulty in studying (1.1)-(1.2) is the non-regularity of the flux, since Du is just bounded and it can not be multiplied by λ, whenever this is a Radon measure.Several approaches have been proposed, by relaxing the Monge-Kantorovich problem (see [7], [18] or [8]).
Recently, this charges approach was extended in [3] to a class of coercive nonlocal problems considered in [25] with fractional gradient constraint of the type (1.4) |D s u| ≤ g, 0 < s < 1, where D s is the distributional fractional Riesz gradient.The fractional s-gradient D s has been recently studied by several authors [27], [28], [13], [14].It may be defined via smooth functions where D s • denotes the s-divergence and (−∆) s the fractional s-Laplacian.For smooth functions with compact support D s can also be equivalently defined by a vector-valued fractional singular integral, which satisfies elementary physical requirements, such as translational and rotational invariances, homogeneity of degree s under isotropic scaling and certain basic continuity properties [28], in order to model long-range forces and nonlocal effects in continuum mechanics.
Another important property of D s is due to the fact that the Riesz kernel I 1−s approaches the identity operator as s → 1, which implies that D s u −→ Du in L p -spaces, provided Du ∈ L p (R d ) = L p (R d ) d (see Section 2, for details).However it should be noted that even when u has compact support in R d and D s u makes sense as a p-integrable function, in general, D s u has not compact support in contrast with Du = D 1 u.
Here we shall be concerned with the more general fractional Monge-Kantorovich-type problem for a function u, satisfying u = 0 in R d \ Ω, and a charge λ, such that |D s u| ≤ g s , λ ≥ 0 and λ(|D s u| − g s ) = 0.
(1.7) s For a bounded positive threshold g s , the first condition in (1.7) s holds a.e.x ∈ R d , for 0 < s < 1, and a.e. in Ω, for s = 1, while the second and third ones are interpreted in L ∞ (R d ) ′ and in L ∞ (Ω) ′ , respectively.
The equation (1.6) s must be interpreted in an appropriate functional space duality with the bilinear form associated to a linear operator for 0 < s ≤ 1, possibly degenerate, in the general form: where the nonnegative matrix A = A(x) has integrable coefficients, which may degenerate or even vanish completely, the vector fields b and d, as well as the function c and the given data f and f are also merely integrable in the case of bounded g s , even in the classical local case s = 1.
The paper is organised as follows: in Section 2 we develop the required functional framework for the Riesz fractional derivatives and we recall and prove some interesting properties of the spaces Λ s,p 0 (Ω), including (1.9); in Section 3, we precise the assumptions on L s , which may be a degenerate operator, and we prove the existence of a solution to the corresponding pseudomonotone variational inequality with the convex set of the s-gradient constraint (1.4) in H s 0 (Ω) and in Λ s,∞ 0 (Ω) for nonnegative threshold g ∈ L 2 loc (R d ) and g ∈ L ∞ loc (R d ), respectively.We also give sufficient conditions for the operator L s to be strictly coercive in H s 0 (Ω) and, as a consequence, we extend the strong continuous dependence (and the uniqueness) of the transport potential u with respect to the data, including the continuous dependence on the s-gradient thresholds.
Our main results are in Section 4, where we prove the existence of a generalised transport potential-density pair solving the Monge-Kantorovich equations (1.6) s and (1.7) s under rather general conditions on the operator L s , including the L 1 integrability of its coefficients.The proof is based on a new generalised weak continuous dependence on the pair (u, λ) with respect not only on the coefficients of L s and on the data f , f (in L 1 ) but also on the threshold g (in L ∞ ) and on the solvability and a priori estimates of a suitable family of approximation problems in the space Λ s,q 0 (Ω), for a large finite q, with a penalisation of the s-gradient and with a nonlinear regularisation of q-power type of the possible degenerate operator L s .Finally in Section 5 we extend the weak convergence on the generalised localisation of the transport potentials and densities as the fractional parameter s → 1, improving the result of [3].In Sections 4 and 5, we work with generalised sequences, also called nets, see for instance [19].

The functional framework
Following [27] we recall that the fractional gradient of order s ∈ (0, 1), denoted by D s = D s 1 , . . ., D s d , may be defined in the distributional sense by , for each i = 1, . . ., d: (2.1) The Riesz kernel of order α ∈ (0, 1), for x ∈ R d \ {0}, is given by , and it satisfies the following well-known properties which proof is reproduced for completeness.
We fix the notation B(x, r) for the open ball centered at x ∈ R d and radius r > 0.
Lemma 2.1.Let I α be the Riesz kernel, 0 < α < 1, p ∈ (1, ∞) and R > 0.Then, denoting by σ d−1 the surface area of the unit sphere in R d , we have: the conclusions follows.□ As a consequence, the Riesz kernel is an approximation of the identity, and it was observed by Kurokawa [20], in the sense that as observed in [25], we have We shall need the following stronger result which is also a consequence of this observation (see also Proposition 2.10 of [20] for the pointwise convergence).
Proof.Let ε > 0 and 0 < δ < 1 be such that Using Lemma 2.1 consider α 0 such that, for 0 < α < α 0 , Then, for all x ∈ R d , B(0,δ) Hence and the conclusion follows.□ As it was proved in [27, Theorem 1.2], the fractional gradient satisfies (2.2) In particular, as an immediate consequence of Theorem 2.2, we obtain the uniform approximation of continuous gradients by their fractional gradients.
Remark 2.4.The convergence in (2.3) has been shown with a different proof for functions in , respectively in Proposition 4.4 and in Theorem 4.11 of [13].This property can be seen as a localization of the fractional gradient.

It has also been shown for functions in
For smooth functions with compact support, as it was observed in [14], the distributional Riesz fractional gradient D s can also be defined for 0 < s < 1 by where µ s = (d + s − 1)γ d,1−s is bounded and lim s→1 µ s = 0.
Let Ω be a bounded open subset of R d and set In this work, for a function u defined in Ω, we still denote its extension by zero to R d by u.
From (2.2) or (2.4), we see that for a function u ∈ C 1 c (Ω), while Du = 0 in R d \ Ω, D s u is in general different from zero in the whole R d .Nevertheless the following remark holds.
For 1 < p < ∞, in [27] it was proved that Λ s,p (R d ) (denoted there as X s,p (R d )) is equal to  [11]).It is worth to recall that denotes the fractional Sobolev-Slobodeckij spaces.In fact, Λ k,p (R d ) = W k,p (R d ) for nonnegative integers k or when p = 2 and s > 0, being Also in [27] it was shown the fractional Sobolev inequality for 1 < p < ∞ and 0 < s < 1, Clearly, considering the smooth functions with compact support trivially extended by zero outside their support, we have Λ s,p 0 (Ω) ⊂ Λ s,p (R d ).We observe that, by definition, for u ∈ Λ s,p 0 (Ω) the by using Fubini's Theorem, and letting n → ∞, we conclude that D s u = D(I 1−s u), i.e.D s u is the distributional Riesz fractional gradient of u.Moreover, in the limit, we may also conclude that u ∈ Λ s,p 0 (Ω) also satisfies (2.7) From (2.5) we obtain a Poincaré inequality for some C p > 0, and in Λ s,p 0 (Ω) we shall use the equivalent norm We can extend the definition (2.6) for p = 1 and define The fractional Poincaré inequality (2.8) can be made more precise with respect to s, 0 < s < 1, to also include the limit cases p = 1 and p = ∞.
Proposition 2.6.Let Ω ⊂ R d be a bounded open set.Then there exists a constant Proof.For 1 < p < ∞, this is Theorem 2.9 of [5], but the same proof is still valid for p = 1.

10). □
In addition, in a bounded open set Ω ⊂ R d satisfying the extension property, it is well known that Λ s,2 0 (Ω) = W s,2 0 (Ω) = H s 0 (Ω) (see, for instance, [22]).Although there is no monotone inclusions in p of L p (R d ) the following result holds.
As a consequence, the following inclusions hold and are continuous, since there exists C p,q > 0 such that In addition, C 1,q = E s , where E is independent of s, and C p,q is independent of s, if p > 1.
We denote the dual space of Λ s,p 0 (Ω) by Λ −s,p ′ (Ω), 0 < s < 1, 1 < p < ∞, and we have a similar characterization in terms of the fractional s-gradient as it was shown in [11, Theorem Proposition 2.13.Let 0 < s < 1, 1 < p < ∞ and F ∈ Λ −s,p ′ (Ω).Then there exist functions ) defines a linear form in Λ s,∞ 0 (Ω).However, these forms do not exhaust Λ s,∞ 0 (Ω) ′ .We shall also work with the dual of L ∞ (R d ), which is also denoted as ba(R d ) (see, for instance, [23] and [2]) and their elements are sometimes called charges.We recall (see Example 5, Section 9, Chapter IV of [29]) that an element λ ∈ L ∞ (R d ) ′ can be represented by a Radon integral (2.20) ⟨λ, φ⟩ = for a finitely additive measure λ * , which is of bounded variation and absolutely continuous with respect to the Lebesgue measure in R d .
Exactly as for the Lebesgue integral, we have the Hölder inequality for positive charges (see [23, p.122]).

Variational inequalities with s-gradient constraints
Let Ω ⊂ R d be open and bounded, with the extension property, i.e., the extension of This holds, in particular, for domains with Lipschitz boundaries (see for instance [15,Section 5]).To consider s-gradient constrained problems, we define the following closed convex sets for prescribed thresholds satisfying or, in the bounded case, In Lemma 3.2, we will see that these assumptions on g are enough for K s g to be bounded in H s 0 (Ω) and Λ s,∞ 0 (Ω), respectively.For 0 < s ≤ 1, we define a bilinear form by letting Here the principal part may be degenerate, under the assumption on the matrix A = A(x): In addition, we shall assume that the coefficients of the bilinear form satisfy, when u, v ∈ H s 0 (Ω), or, in the bounded case, when u, v ∈ Λ s,∞ 0 (Ω), Similarly, for 0 < s ≤ 1, we may define the linear form where, by the Sobolev embedding (2.5), 2 # = 2d d+2s if 0 < s < d 2 , or 2 # = q for any q > 1 when s = 1  2 , and 2 # = 1 when 1 2 < s ≤ 1 or, in the bounded case, for any v ∈ Λ s,∞ 0 (Ω), with Notice that in the case s = 1, since u, v and Du, Dv are zero in R d \Ω, all the integration domains in (3.4) and (3.8) reduce to Ω.Then, for 0 < s ≤ 1, there exists a solution of the s-gradient constraint variational inequality We will use the following lemma in the proof of this theorem.
Lemma 3.2.For 1 ≤ p ≤ ∞ and g ∈ L p loc (R d ), with g ≥ 0, the set K s g is bounded in Λ s,p 0 (Ω).More precisely, there exists R = R(p, s) such that, for u ∈ K s g , Proof.By the Theorem 2.7, when p < ∞, choosing R such that , and using (2.11), we have from where we obtain the first inequality.Letting p → ∞, the second inequality follows.□ Proof.(of Theorem 3.1) This existence result in the Hilbertian case i) is a consequence of a theorem of H. Brézis (see [9] or [21, Theorem 8.1, p. 245]), since K s g is a nonempty, closed and bounded convex set of H s 0 (Ω) and the operator P : Indeed, taking u n −⇀ n u in H s 0 (Ω), which by compactness of the embedding (2.15) (for p = 2 and 1 ≤ q < 2 * = 2d d−2s if 2s < d, for all q ≥ 1 if 2s = d and for q = ∞ if 2s > d), we may assume also that u n → n u in L q (Ω).Write P in the form It is then clear that the assumptions (3.7) and (3.10) imply ), and we have Hence (3.14) follows easily by noting that the assumption (3.5) implies , and hence it suffices to take the limit inferior in In the non-Hilbertian case, we start by approximating the data, in the respective spaces, by smooth functions with compact support A m , b m , d m , c m , f #m and f m and we let u m be a solution of the variational inequality (3.11) with these data, which exists by the previous case.
) by Lemma 3.2, using the compact embedding (2.16), there )-weak, for any p < ∞, in particular for p = p ′ 1 and p = q ′ 1 .Using the above convergences, we immediately have, for any v ∈ K s g , On the other hand, using the monotonicity of A m , for any v ∈ K s g we have Choosing v = u + t(w − u) ∈ K s g , for t ∈ (0, 1), as test function, we obtain after letting t −→ 0 + .Therefore u solves (3.11).□ Remark 3.3.To obtain the uniqueness to (3.11) it suffices to require the strict positivity of the bilinear form which needs stronger assumptions on its coefficients.
Remark 3.4.The constrained problem (3.11) for u ∈ K s g determines the existence of an element Γ = Γ(u) ∈ H −s (Ω) belonging to the sub-differential of the indicatrix function [21, p.203]), which is given by where L s : K s g −→ H −s (Ω) is the linear operator defined by the bilinear form as in (3.4).A main question is to relate Γ to the solution u, for instance trough the existence of a Lagrange multiplier λ such that Γ = λD s u.This has been shown only in very special cases with the classical gradient (s = 1) (see [10] and [24], for more references).
The existence result of Theorem 3.1 includes the degenerate case A ≡ 0 in (3.11).On the other, when the matrix A is strictly elliptic, i.e., if we replace (3.5) by assuming the existence of we may give the following sufficient condition for the bilinear form (3.4) to be strictly coercive, by imposing Here C * is the Sobolev constant of the embedding 16) and (3.17) for 2s < d, and g ∈ L 2 loc (Ω).Then, for any f # and f satisfying (3.9), there exists a unique solution to (3.11).If u denotes the solution to (3.11) for f # and f , we have Proof.Using Hölder and Sobolev inequalities, we have that, for v ∈ H s 0 (Ω), and Therefore, using (3.16) and (3.17), we obtain ) and Stampacchia's theorem immediately yields the existence and uniqueness of the solution to (3.11).
Taking v = u in (3.11) for u and v = u in (3.11) for u, and using (3.19), we obtain and (3.18) easily follows.□ Remark 3.6.We observe that the assumption (3.6) is slightly stronger than the integrability conditions in Theorem 3.5 for the case 2s < d, including s = 1.However, for s ≥ d 2 , we may have the assumption (3.6) with any r > 1, when s = 1, d = 2, and even b, d, c ∈ L 1 when s ≥ 1 2 , d = 1, with the respective norms in the assumption (3.17).Note that Theorem 3.5 extends Theorem 2.1 of [3], in which the coefficients b, d and c are zero.
The coercivity assumption (3.17 where the positive constants C 1 and C ′ 1 depend on δ, g * , g * , d, s, Ω and linearly on the where C 0 > 0 is the Poincaré constant in (2.10). Set Observe that u i j = µu j ∈ K s g i (i ̸ = j, i, j = 1, 2) and so it can be used as test function in (3.11) for L s i and f #i , f i .For i = 1, 2, we obtain or equivalently, Since u i j − u j = (µ − 1)u j and 0 ≤ 1 − µ ≤ η g * , setting M = max{g * , C 0 s g * }, we may estimate the middle term of (3.22) by and the last one by Setting w = u 1 − u 2 and by using (3.22) for i = 2, we have where Summing (3.22) for i = 1 with (3.23) and using the coercivity (3.19), we obtain where To conclude (3.21) it suffices to use the continuous embedding (2.16), which guarantees the existence of a constant C β > 0 and 0

□
The following proposition shows that we can replace the assumption (3.20) by (3.20) loc in the above theorem.
Proposition 3.8.Let g ∈ L ∞ loc (R d ) be positively lower bounded in any compact set and such that Then there exists h ∈ L ∞ (R d ), with positive lower bound in R d and such that K s h = K s g .More precisely, we can choose Proof.Using remarks 2.5, 2.9 and the inequality (3.12) for p = 1, there exists R 0 = R 0 (s) > 0, independent of g, such that, for R ≥ R 0 , and u ∈ K s g , we have Then h satisfies assumption (3.20) and and then, using (3.25), |D s u(x)| ≤ g(x).□ Remark 3.9.Assuming only (3.20) loc , Theorem 3.7 remains valid by taking R > 0 sufficiently large and replacing This is true because in the proof of the last proposition for g 1 and g 2 , we can define h 1 and In particular, with these assumptions we also have uniqueness of solution for the variational inequality.

Transport potentials and densities
In this section we consider the Lagrange multiplier problem for 0 < s ≤ 1, associated with bounded s-gradient constraints: find the generalised transport potential-density pair (u, λ) ∈ In the case s = 1 the solution (u, λ) is to be found in W 1,∞ 0 (Ω)×L ∞ (Ω) ′ and the test functions , and u solves the variational inequality (3.11).
The proof of this existence theorem is obtained by a suitable penalisation of the s-gradient constraint, combined with an elliptic nonlinear regularisation, and by a weak stability property of the generalised formulation (4.1) given by the following theorem.
Taking v = u ε in (4.20) and using (4.28) and the semicontinuity (4.14 with (recall that k and, afterwards from (4.29), also From (4.24) it follows ⟨λ, |D s u| 2 − g 2 ⟩ = 0 and we conclude, as in (4.18), that The general case follows by Theorem 4.2, by approximating with solutions of (4.1a) ν , (4.1b) ν with data A ν , b ν , d ν , c, f # , f ν and g satisfying (3.5), (4.19) and (3.20) and converging strongly in L p 1 , L 1 and L q 1 , for instance by using f , where χ B(0, 1 ν ) denotes the characteristic function of B(0, 1 ν ).Finally, since u ∈ K s g , given v ∈ K s g , taking v − u as test function in (4.1a) and noting that it is clear that u solves the variational inequality (3.11), which concludes the proof of Theorem  Here we can also allow a variable threshold g s under the assumption Proof.We adapt the steps of the proof of Theorem 4.2: i) a priori estimates with respect to s; ii) existence of limits of generalised sequences, by compactness, and iii) characterization of those limits as solutions of the local problem (5.1) 1 -(5.2) 1 .For 0 < σ < s < 1, using the Poincaré inequality (2.8), we have σ C 0 ∥u s ∥ L ∞ (Ω) ≤ ∥D s u s ∥ L ∞ (R d ) .Then by the assumption (5.3) we obtain for any u s ∈ K s gs solution of (5.1) s -(5.2) s , we get that ≤ p < ∞, ∥u s ∥ C 0,β (Ω) ≤ C β , for any β, 0 < β < s < 1, (5.6) where the constant C β is independent of s.
Letting Ψ s = λ s D s u s , arguing exactly as in (4.5)-(4.7)and (4.8), by replacing the label ν by s, we obtain that where C 1 > 0 is a constant independent of s, σ < s < 1 Therefore, by compactness, in particular, by (5.6) and (2.18), there are u ∈ C 0,α (Ω) ∩ Λ σ,p (Ω), for 0 < α < β, 0 < σ < 1 and Observe that we still have the lower semicontinuity property by the convolution of the classical gradient with the Riesz kernel I 1−s , i.e., D s u = I 1−s * Du = D(I 1−s * u), with the nice properties (−∆) s u = −D s • (D s u) and (1.5) for a constant C * > 0, where p * = dp d−sp , if sp < d, as well as the fractional Trudinger (p * < ∞, if sp = d) and Morrey (p * = ∞, if sp > d) inequalities.If sp > d, in the left side of (2.5), we may take the semi-norm of β-Hölder continuous functions, 0 < β = s − d p .For an open bounded set Ω ⊂ R d , we define the subspace, for 0 < s ≤ 1, and for q = ∞ if sp > d, the embeddings being compact in the case sp < d only for q < dp d−sp = p * .Also the embeddings (2.16) Λ s,p 0 (Ω) ⊂ C 0,β (Ω), for sp > d are continuous for 0 < β ≤ s − d p and compact for 0 < β < s − d p , where C 0,β (Ω) denotes the space of Hölder continuous functions in Ω of exponent β.Consequently, by Proposition 2.8, we have the compact embeddings
) in the case of a bounded threshold of the s-gradient, under the stronger assumptions (3.20) 0 < g * ≤ g(x) ≤ g * for a.e.x ∈ R d , also yields strong continuous dependence of the solutions of (3.11) with respect to the variation of the coefficients of L s , of the data and of the threshold g.In fact, the assumption (3.20) can be weakened as follows (3.20) loc g ∈ L ∞ loc (R d ) with positive lower bound in any compact and lim |x|→∞ g(x)|x| d+s = ∞, as it will be shown in Proposition 3.8.Theorem 3.7.Let u i denote the solution of (3.11) corresponding to the data A i , b i , d i , c i ,

h 2
as in (3.26) with the same R and h 1 equal to h 2 outside Ω R .Remark 3.10.If we assume (3.5), b = d = 0 and c ≥ c * > 0 instead (3.17), keeping the other assumptions in Theorem 3.7, we still have a weaker continuous dependence result, replacing

4. 1 . □ Remark 4 . 3 .
Theorem 4.2, as a weak continuous dependence result, generalises Theorem 3.5 to the case of degenerate operators, including the case A ≡ 0, with L 1 -data.In fact, if A satisfies (3.5) and (3.6) holds with the strictly coercive assumption (3.17), it is clear that u solving problem (4.1) is unique.On the other hand, the uniqueness of λ is an open problem, even in the local case of s = 1, which was considered first for the Laplacian in[4] in the special casef # ∈ L 2 (Ω) and f = 0.Remark 4.4.As in Section 3, we may assume in Theorems 4.1 and 4.2 that g and g ν satisfy ((3.20) loc ) instead (3.20), with uniform limit in ν.
although that proof is equally valid for functions only inC 1 c (R d ), see [14, Proposition 2.2].As a consequence of well-known properties of the Riesz potential, (2.2) is then also valid for functions u in the usual Sobolev space W1,p where g s are the Bessel potentials, for s ∈ R,