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 → ∞ p, of the following minimization problem: min ⁡ 1 p ⁢ ∫ Ω | ∇ ⁡ v | p + 1 2 ⁢ ∫ Ω (v - f) 2, v ∈ W 1, p ⁢ (Ω). \1{p_| v|^p+12_ (% v-f) ^2, v W^1, p () \}. We show that the limit problem provides the best approximation, in the L 2 L^{2} -norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an 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) (N-1) -Hausdorff measure of the jump set of the function.