The best approximation of a given function in $L^2$-norm by Lipschitz functions with gradient constraint

TL;DR

This paper investigates the asymptotic behavior of a minimization problem to find the best Lipschitz approximation of a function in the $L^2$-norm, revealing a PDE characterization and extending to related nonvariational problems and jump set measures.

Contribution

It introduces a new asymptotic analysis of a variational problem leading to the best Lipschitz approximation in $L^2$-norm with PDE characterization, including nonvariational extensions.

Findings

Limit problem yields the best Lipschitz approximation in $L^2$-norm.

The approximation satisfies a PDE in the viscosity sense.

Extension to nonvariational equations and jump set measures.

Abstract

The starting point of this paper is the study of the asymptotic behavior, as $p\to\infty$, of the following minimization problem $$ \min\left\{\frac1{p}\int|\nabla v|^{p}+\frac12\int(v-f)^2 \,, \quad \ v\in W^{1,p} (\Omega)\right\}. $$ We show that the limit problem provides the best approximation, in the $L^2$-norm, of the datum $f$ among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover such approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the $(N-1)$-Hausdorff measure of the jump set of the function.