Lagrange multipliers and transport densities

Abstract

In this paper we consider a stationary variational inequality with nonconstant gradient constraint and we prove the existence of solution of a Lagrange multiplier, assuming that the bounded open not necessarily convex set Ω has a smooth boundary. If the gradient constraint g is sufficiently smooth and satisfies ∆g 2 ≤ 0 and the source term belongs to L ∞ (Ω), we are able to prove that the Lagrange multiplier belongs to L q (Ω), for 1 < q < ∞, even in a very degenerate case. Fixing q ≥ 2, the result is still true if ∆g 2 is bounded from above by a positive sufficiently small constant that depends on Ω, q, min Ω g and max Ω g. Without the restriction on the sign of ∆g 2 we are still able to find a Lagrange multiplier, now belonging to L ∞ (Ω). We also prove that if we consider the variational inequality with coercivity constant δ and constraint g, then the family of solutions (λ δ , u δ) δ>0 of our problem has a subsequence that converges weakly to (λ 0 , u 0), which solves the transport equation. Résumé Dans cet article, nous considérons une inégalité variationnelle stationnaire avec une restriction non-constante sur le gradient et nous prouvons l'existence d'un multiplicateur de Lagrange, en supposant que l'ensemble ouvert et borné Ω, pas nécessairement convexe, a une frontière régulière. Si la restriction du gradient g est suffisamment régulière et satisfait ∆g 2 ≤ 0 et le terme source appartient a L ∞ (Ω), nous pouvons prouver que le multiplicateur de Lagrange appartientà L q (Ω), pour 1 < q < ∞, même dans un cas très dégénéré. Si nous fixons q ≥ 2, le résultat est aussi vrai si ∆g 2 est borné par une constante positive et suffisamment petite qui dépend de Ω, q, min Ω g et max Ω g. Sans la restriction sur le signe de ∆g 2 nous sommes capables de trouver un multiplicateur de Lagrange, maintenant appartenantà L ∞ (Ω). Nous montrons aussi que si l'on considère l'inégalité variationnelle avec la coercitivité constante δ et la restriction g, alors la famille des solutions (λ δ , u δ) δ>0 de notre problème a une sous-suite qui converge faiblement vers (λ 0 , u 0), ce qui résout l'équation de transport.

On a class of nonlocal problems with fractional gradient constraint

“…to the classical problem for D. We remark that, in this case, our results are new for data in L 1 and the general elliptic operator −D • (AD ), extending [3] where the charges approach was introduced for −∆ with f # ∈ L 2 (Ω) and f = 0. For a recent survey on gradient type constrained problems see [11].…”

“…which is even continuous if Ω is convex (see [11] for references). Although (2.14) is an open question in the general case of Theorem 2.1, for strictly positive bounded threshold g, it has been shown to hold in the sense of finite additive measures in [10], following the case σ = 1 of [3].…”

Variational and Quasi-Variational Inequalities with Gradient Type Constraints

“…In fact, arguing as in [8,Theorem 1.3], which corresponds only to the particular scalar case L = ∇, we can prove the following theorem:…”

“…(Ω ) and in [8] this result was extended for δ ≥ 0, with f ∈ L ∞ (Ω ) and g only in L ∞ (Ω ), as it is stated in the theorem above, but for L= ∇. Besides, when g ∈ C 2 (Ω ) and ∆ g 2 ≤ 0, in [8] it is also shown that λ δ ∈ L q (Ω ), for any 1 ≤ q < ∞ and δ ≥ 0.…”

“…We set X 2 = w w w ∈ H 1 0 (Ω ) d : ∇ · w w w = 0 , d = 2, 3, which is an Hilbert space for Dw w w L 2 (Ω ) d 2 compactly embedded in H = w w w ∈ L 2 (Ω ) d : ∇ · w w w = 0 . Defining K G[u u u](t) for each t ∈ (0, T ) by (8) and giving f f f ∈ L 2 (Q T ) d and u u u 0 ∈ X 2 satisfying (106) at t = 0, i.e. u u u 0 ∈ K G[u u u 0 ] , in order to apply Theorem 21, we set…”