Equivalence between the growth of \(\int_{b(x,r)}| \nabla u| ^ pdy\) and T in the equation \(P[u]=T\)
type: Item
instance of: scholarly article
title: Equivalence between the growth of \(\int_{b(x,r)}| \nabla u| ^ pdy\) and T in the equation \(P[u]=T\)
zbMATH Open document ID: 0707.35033
DOI: 10.1016/0022-0396(90)90043-O
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: Let us define \[ (I)\;M_{\lambda}^{-1,q}(\Omega)=\{T\in W^{- 1,q}(\Omega)\text{ such that }\sup_{k}[\rho^{-(\lambda /q)}\| T\|_{\bar W^{-1,q}}(\Omega (x,\rho))]<\infty \}, \] where \(\Omega (x,\rho)=\Omega \cap B(x,\rho)\), \(K=\{(\rho,x)\in I\times \Omega \}\), \(I=(0\), diameter of \(\Omega\)), \[ (II)\;Au+F(u,\nabla u)=T, \] where (i) \(\Omega\) is an open set in \({\mathbb{R}}^ N\), (ii) for each \(u\in W^{1,p}_{loc}(\Omega)\cap L^{\infty}_{loc}(\Omega)\), \(Au=- \sum^{N}_{i=1}(\partial /\partial x_ i)a_ i\) (x,u,\(\nabla u)\), \(a_ i\) are Borelian functions from \(\Omega \times {\mathbb{R}}\times {\mathbb{R}}^ N\) into R, and for a.e. \(x\in \Omega\) and all \((\eta,\xi)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), \[ | a_ i(x,\eta,\xi)| \leq a(| \eta |)[| \xi |^{p-1}+a_ 0(x)],\;a_ 0\in L_{loc}^{q,N-p+\beta}(\Omega),\;\beta >0 \] and a is an increasing function \(R_+\to R_+\) and \(a_ i\) satisfies some monotonicity and coerciveness, and (iii) the nonlinearity F is such that for a.e. \(x\in \Omega\), \(\forall (\eta,\xi)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), \[ | F(x,\eta,\xi)| \leq f(| \eta |)[| \xi |^{p- \gamma}+f_ 0(x)], \] where \(\gamma >0\), \(f_ 0\in L_{loc}^{1,N- p+\sigma}(\Omega)\), \(\sigma >0\) and f is an increasing function \({\mathbb{R}}_+\to {\mathbb{R}}_+.\) If u is a local solution of \(Au+F(u,\nabla u)=T\), then \(T\in M^{- 1,q}_{\lambda,loc}(\Omega)\) for \(\lambda >N-p\), \(1/p+1/q=1\) if and only if for all subsets \(\Omega'\) relatively compact, there exist \(C>0\) and \(\sigma>N-p\) such that \(\forall x\in \Omega'\), \(\forall R>0\); \(B(x,2R)\subset \Omega'_ 0\) we have \[ \int_{B(x,R)}| \nabla u|^ p dy\leq C\cdot R^{\sigma}. \]
reviewed by: Y.Suyama
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 4161199
zbMATH Keywords: Morrey spaces
zbMATH Keywords: monotonicity
zbMATH Keywords: coerciveness
author: J.-M. Rakotoson
MaRDI profile type: Publication
cites work: Q5624209
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cites work: Q3252874
cites work: Harnack inequalities for quasi-minima of variational integrals
cites work: Behavior at the boundary of solutions of quasilinear elliptic equations
cites work: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105)
cites work: Q5187597
cites work: On some non-linear elliptic differential functional equations
cites work: Thin sets in nonlinear potential theory
cites work: Q5539922
cites work: Linear and quasilinear elliptic equations
cites work: Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder
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cites work: Q5890172
cites work: Q5518064
cites work: A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations
cites work: Q1107021
cites work: On some degenerate and nondegenerate quasilinear elliptic systems with nonhomogeneous Dirichlet boundary condition
cites work: Q3745373
cites work: Local Behavior of Solutions of Quasilinear Elliptic Equations With General Structure
cites work: Solutions to the obstacle problem as Green potentials
cites work: Q5735444
cites work: Local behavior of solutions of quasi-linear equations
cites work: On harnack type inequalities and their application to quasilinear elliptic equations
full work available at URL: 0022-0396(90)90043-o
OpenAlex ID: W2018817282
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description: scientific article; zbMATH DE number 4161199