Approximation of Lipschitz Functions Preserving Boundary Values

Given an open subset of a Banach space and a Lipschitz real-valued function defined on its closure, we study whether it is possible to approximate this function uniformly by Lipschitz functions having the same Lipschitz constant and preserving the values of the initial function on the boundary of the open set, and which are k times continuously differentiable on the open. A consequence of our result is that every 1-Lipschitz function defined on the closure of an open subset of a finite-dimensional normed space of dimension greater than one, and such that the Lipschitz constant of its restriction to the boundary is less than 1, can be uniformly approximated by differentiable 1-Lipschitz functions preserving the values of the initial function on the boundary of the open set, and such that its derivative has norm one almost everywhere on the open. This result does not hold in general without assumption on the Lipschitz constant of the restriction of the initial function to the boundary.