Measure data problems, lower-order terms and interpolation effects
type: Item
instance of: scholarly article
title: Measure data problems, lower-order terms and interpolation effects
zbMATH Open document ID: 1305.35057
author: Agnese Di Castro
author: Giampiero Palatucci
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2014-06-18T00:00:00Z
review text: The nonlinear PDE \[ -\mathrm{div} \;a\left(x,Du\right)+g(x,u)=f \] is studied in an open bounded subset \(\Omega\) of \(\mathbb{R}^n\), \(n\geq2\) with Dirichlet boundary condition \(u=0\) on \(\partial \Omega\). Here \(f\in L^1(\Omega)\), the vector field \(a:\Omega\times\mathbb{R}^n\rightarrow\mathbb{R}^n\) is a Carathéodory regular and statisfies the following monotonicity and \(p\)-growth conditions: \[ \nu(s^2+\left|z_1\right|^2+\left|z_2\right|^2)^{(p-2)/2}\left|z_2-z_1\right|^2\leq\left\langle a(x,z_2)-a(x,z_1),z_2-z_1\right\rangle \] and \[ \left|a(x,z)\right|\leq L\left(s^2+\left|z\right|^2\right)^{(p-1)/2} \] for every \(z_1,z_2,z\in\mathbb{R}^n\) and \(x\in\Omega\); the constants \(\nu\), \(L\) and \(s\) are s.t. \(0<\nu\leq1\leq L\) and \(s\geq0\). It is also assumed that the function \(g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}\) is a Carathéodory one such that for all \(t\) and a.e. \(x\in\Omega\) the inequality \(g(x,t)\mathrm{sgn}(t)\geq\alpha_0\left|t\right|^m\) holds for some positive constants \(\alpha_0\) and \(m\), and \(\sup_{\left|t\right|\leq\tau}g(\cdot,t)\in L^1_{\mathrm{loc}}(\Omega)\). A general approach to the regularity of solutions is suggested in the paper. It is based on estimates on the level sets of the maximal operator of the gradient of solutions in terms of the level sets of the datum \(f\). They imply local estimates in the most familiar function spaces (Lebesgue, Orlicz, Lorentz--Morrey). The results obtained are in clear accordance with the classical Gagliardo-Nirenberg interpolation theory.
reviewed by: Tihomir Gyulov
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6305037
zbMATH Keywords: nonlinear elliptic equation
zbMATH Keywords: Morrey-Lorentz regularity
zbMATH Keywords: rearrangement function spaces
zbMATH Keywords: Gagliardo-Nirenberg interpolation inequalities
MaRDI profile type: Publication
full work available at URL: s10231-012-0277-7
OpenAlex ID: W2003886140
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DOI: 10.1007/S10231-012-0277-7
description: scientific article; zbMATH DE number 6305037