  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.
