Buccheri, StefanoLeonori, TommasoRossi, Julio D.2024-11-082024-11-082024-04-24Buccheri, Stefano, Leonori, Tommaso and Rossi, Julio D.. "The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint" Advances in Calculus of Variations, 2024. https://doi.org/10.1515/acv-2023-00581864-8266https://doi.org/10.1515/acv-2023-0058https://hdl.handle.net/20.500.14468/24318The starting point of this paper is the study of the asymptotic behavior, as p → ∞, of the following minimization problem: min{1 p ∫ Ω |∇v|p + 1 2 ∫ (v − f)2, v ∈ W1,p(Ω)}. Ω We show that the limit problem provides the best approximation, in the L2-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)-Hausdorff measure of the jump set of the function.eninfo:eu-repo/semantics/openAccess12 Matemáticasartículop-Laplacianinfinity-LaplacianLipschitz approximations