Existence and Approximation for Variational Problems Under Uniform Constraints on the Gradient by Power Penalty

Abstract

Variational problems under uniform quasiconvex constraints on the gradient are studied. In particular, existence of solutions to such problems is proved as well as existence of lagrange multipliers associated to the uniform constraint. They are shown to satisfy an Euler-Lagrange equation and a complementarity property. Our technique consists in approximating the original problem by a one-parameter family of smooth unconstrained optimization problems. Numerical experiments confirm the ability of our method to accurately compute solutions and Lagrange multipliers. 2. Statement of the problem and main results. Let Ω be a bounded domain in R N with N ≥ 1 and T : Ω × R m×N → [0, ∞[ a Carathéodory function. Let s ≥ 1 and consider a functional J : W 1,s (Ω; R m ) → R ∪ {+∞}, which is supposed to be bounded from below and sequentially lower semicontinuous in the weak topology of W 1,s (Ω; R m ). We are interested in the minimization problem inf{J(v) | T (·, ∇v) ∞,Ω ≤ 1, v ∈ g + W 1,s 0 (Ω; R m )}, (2.1)where T (·, ∇v) ∞,Ω = ess-sup{T (x, ∇v(x)) | x ∈ Ω}, and g ∈ W 1,∞ (Ω; R m ) ∩ C(Ω; R m ) is a given function satisfying J(g) < +∞ and T (x, ∇g(x)) ≤ 1 for a.e. x ∈ Ω.Then (2.1) may be rewritten as inf J ∞ (v) | v ∈ g + W 1,s 0 (Ω; R m ) .(2.3)

Lagrange multipliers and transport densities

“…The existence of λ ∈ M (Ω) was proved in [5] (for more general operators), assuming that Ω is convex and f ∈ L q (Ω), with q > d. Here M (Ω) denotes the set of Radon measures. A recent result in this direction can be found in [1]. The existence of an essentially bounded Lagrange multiplier was proved, in [21], for the parabolic version of problem (2), with nonhomogeneous Dirichlet boundary condition and in [22] for a constraint g such that ∆g 2 ≤ 0 and homogeneous Dirichlet boundary condition, in both cases with f depending only on the variable t.…”