Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient
type: Item
instance of: scholarly article
title: Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient
zbMATH Open document ID: 1359.35077
author: I. Birindelli
author: F. Demengel
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2017-01-11T00:00:00Z
review text: This paper deals with the regularity up to the boundary of bounded viscosity solutions \(u\) to the nonhomogeneous Dirichlet problem associated to the nonlinear elliptic equation \(|\nabla u|^\alpha F(D^2u) +|\nabla u|^\beta b=f\) in a bounded \(C^2\) domain \(\Omega\) of \(\mathbb{R}^N\). The authors establish that any bounded viscosity solution \(u\) belongs to \(C^{1,\gamma}(\overline{\Omega})\) for \(b,f\in C(\overline{\Omega})\), the function \(F\) being uniformly elliptic, the boundary datum belonging to \(C^{1,\gamma_0}(\partial\Omega)\) and \(0<\beta<1+\alpha\) and \(\alpha\geq 0\). The proof relies on the localization to \(B(x,r)\cap \{y_N\geq a(y')\}\) for any \(x\in B \cap \{y_N> a(y')\}\) with \(B(x,1)\subset B\) and any \(r<1\), by making recourse to the contradiction and recursive arguments. Also the Hölder regularity of the gradient is discussed for bounded viscosity solutions \(u\) to the Dirichlet problem associated to the generalized nonlinear elliptic equation \(|\nabla u|^\alpha F(D^2u) +\sum_{i=1}^{k} |\nabla u|^{\beta_i} b_i=f \) in \(B(0,1)\). The proof appeals to the contradiction argument once more.
reviewed by: Luisa Consiglieri
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D40
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 6673667
zbMATH Keywords: regularity
zbMATH Keywords: fully nonlinear equations
zbMATH Keywords: viscosity solution
MaRDI profile type: Publication
full work available at URL: s00030-016-0392-z
OpenAlex ID: W2469341043
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DOI: 10.1007/S00030-016-0392-Z
description: scientific article; zbMATH DE number 6673667