Lagrange multipliers and characteristic functions

TL;DR

This paper analyzes a variational inequality with gradient constraints and obstacles, demonstrating how Lagrange multipliers and characteristic functions describe the problem and establishing solution stability under data convergence.

Contribution

It introduces a novel description of the variational inequality using Lagrange multipliers and characteristic functions, and proves the stability of solutions with respect to data changes.

Findings

Lagrange multipliers encode contact set information.

Characteristic functions are defined on the contact set.

Solutions are stable under convergent data sequences.

Abstract

We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains information about the contact set of the modulus of the gradient of the solution with the gradient constraint, and the characteristic function is defined in the contact set of the solution with the obstacle. Moreover, given a convergent sequence of data, we prove the stability of the corresponding solutions.