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The best approximation of a given function in L2-norm by Lipschitz functions with gradient constraint

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2024-04-24
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info:eu-repo/semantics/openAccess
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De Gruyter
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The 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.
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p-Laplacian, infinity-Laplacian, Lipschitz approximations
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Buccheri, 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-2023-0058
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Facultad de Ciencias
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Matemáticas Fundamentales
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