Second order regularity for degenerate nonlinear elliptic equations
type: Item
instance of: scholarly article
title: Second order regularity for degenerate nonlinear elliptic equations
zbMATH Open document ID: 1401.35106
author: Annamaria Canino
author: Elisa De Giorgio
author: Berardino Sciunzi
published in: Discrete and Continuous Dynamical Systems
publication date: 2018-08-16T00:00:00Z
review text: The authors study the second regularity of weak solutions of degenerate nonlinear elliptic equations of the form \[ -\text{div}\left(A\left(\left |\nabla u\right|\right)\nabla u\right)+b(x)\left| \nabla u\right|^q=f(x)\quad\text{in }\Omega, \] where \(\Omega\subseteq \mathbb{R}^N\) is an open set. The real function \(A: \mathbb{R}^+ \to \mathbb{R}^+\) is of class \(C^1\) such that \[ \text{lim\,sup}_{t\to 0^+} tA(t)<\infty \] and \[ -1<\inf_{t>0}\frac{tA'(t)}{A(t)}:=m_A \leq M_A:= \sup_{t>0} \frac{tA'(t)}{A(t)}<\infty. \] Furthermore, it is assumed that \[ A(s) \geq Ks^{\tilde{\vartheta}} \quad\text{for some }\tilde{\vartheta}\geq 0 \text{ and for all }s>0. \] When \(b(x)\) is not identically zero, then \(q>\frac{\tilde{\vartheta}+1}{2}\). The main aim of the authors is to study the summability of the second derivatives of the solutions which is stated in several theorems under different conditions.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6919416
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: regularity theory
MaRDI profile type: Publication
full work available at URL: dcds.2018184
OpenAlex ID: W2805699843
cites work: Local \({W^{2,m(\cdot)}_{loc}}\) regularity for \(p\)(.)-Laplace equations
cites work: Regularity, monotonicity and symmetry of positive solutions of \(m\)-Laplace equations
cites work: C1 + α local regularity of weak solutions of degenerate elliptic equations
cites work: Universal potential estimates
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: A regularity result for the \(p\)-Laplacian near uniform ellipticity
cites work: Regularity of minima: an invitation to the dark side of the calculus of variations.
cites work: Q5428774
cites work: Some results on the qualitative properties of positive solutions of quasilinear elliptic equations
cites work: Regularity and comparison principles for p-Laplace equations with vanishing source term
cites work: Regularity for quasilinear equations on degenerate singular sets
cites work: Regularity for a more general class of quasilinear equations
Wikidata QID: Q129699326
DOI: 10.3934/DCDS.2018184
description: scientific article; zbMATH DE number 6919416