Existence and Approximability Results for variational problems under uniform constraints on the gradient by power penalty

TL;DR

This paper investigates variational problems with uniform gradient constraints, proving existence of solutions and Lagrange multipliers, and introduces an approximation method with numerical validation.

Contribution

It provides new existence results for solutions and Lagrange multipliers in constrained variational problems and proposes an effective approximation technique.

Findings

Existence of solutions under uniform quasiconvex constraints

Existence of Lagrange multipliers satisfying Euler-Lagrange and complementarity

Numerical experiments confirm the accuracy of the approximation method

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.