Approximation of Lipschitz functions preserving boundary values

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