Approximation of Lipschitz functions preserving boundary values

TL;DR

This paper investigates the approximation of Lipschitz functions on Banach space subsets by smooth functions that preserve boundary values and Lipschitz constants, with specific results for finite-dimensional spaces and boundary conditions.

Contribution

It establishes conditions under which Lipschitz functions can be uniformly approximated by smooth functions preserving boundary values and Lipschitz constants, especially in finite-dimensional spaces.

Findings

Approximation of Lipschitz functions by smooth functions with boundary value preservation.

Conditions for approximation when the boundary Lipschitz constant is less than 1.

Counterexamples showing the necessity of boundary conditions.

Abstract

Given an open subset $\Omega$ of a Banach space and a Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $\Omega$ by $C^k$-smooth Lipschitz functions which coincide with $u_0$ on the boundary $\partial \Omega$ of $\Omega$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$-Lipschitz function $u_0: \overline{\Omega} \to \mathbb{R},$ defined on the closure $\overline{\Omega}$ of an open subset $\Omega$ of a finite dimensional normed space of dimension $n \geq 2$, and such that the Lipschitz constant of the restriction of $u_0$ to the boundary of $\Omega$ is less than $1$, can be uniformly approximated by differentiable $1$-Lipschitz functions $w$ which coincide with $u_0$ on $\partial \Omega$ and satisfy the equation $\| D w\|_* =1$ almost everywhere on $\Omega.$ This result does not hold in general without assumption on the restriction of $u_0$ to the boundary of $\Omega$.