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

Abstract

The starting point of this paper is the study of the asymptotic behavior, as $p\to \infty$, of the following minimization problem $$\min \left\{\frac{1}{p}\int |\nabla v{|}^p+\frac{1}{2}\int (v-f{)}^2,v\in W^{1,p}(\Omega )\right\}.$$ 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.