Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients
type: Item
instance of: scholarly article
title: Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients
zbMATH Open document ID: 1271.42020
author: Vagif S. Guliyev
author: Lubomira Softova
published in: Potential Analysis
publication date: 2013-05-06T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) and let \(\varphi:\Omega\times\mathbb{R}^+\rightarrow \mathbb{R}^+\) be a measurable function. The generalized Morrey space \(M_{p,\varphi}(\Omega)\), \(1\leq p<\infty\), consists of all \(f\in L^p(\Omega)\) such that \[ \|f\|_{M_{p,\varphi}(\Omega)}:=\sup_{x\in\Omega, r>0}\varphi(x,r)r^{-1}\Big(\int_{\Omega_r}|f(y)|^p\,dy\Big)^{1/p}<\infty. \] Here, \(\Omega_r=\Omega\cap\mathcal{B}_r(x)\), where \(\mathcal{B}_r\) is a ball in \(\mathbb{R}^n\) centered in some point \(x\in \Omega\). The generalized Sobolev-Morrey space \(W_{2,p,\varphi}(\Omega)\) is the set of all Sobolev functions \(f\in W_{2,p}(\Omega)\) with distributional derivatives \(D^sf\in M_{p,\varphi}(\Omega)\), endowed with the norm \[ \|f\|_{W_{2,p,\varphi}(\Omega)}=\sum_{0\leq |s|\leq2}\|D^sf\|_{M_{p,\varphi}(\Omega)}. \] Consider the Dirichlet problem for a linear uniformly elliptic operator \[ \begin{cases} \mathcal{L}u:=a^{ij}(x)D_{ij}u=f(x), \quad \text{ a.a. } x\in \Omega\\ u\in W_{2,p,\varphi}(\Omega)\cap W_{1,p}^0(\Omega), \quad p\in (1,\infty) \end{cases} \] where the coefficients \(a^{ij}\) belong to VMO\((\Omega)\), the class of functions of vanishing mean oscillation. This class contains also discontinuous functions. The authors prove that for any function \(f\in M_{p,\varphi}(\Omega)\), \(1<p<\infty\), there exists a unique solution of the Dirichlet problem above and it has second order derivatives in \(M_{p,\varphi}(\Omega)\). Besides \[ \|D^2u\|_{M_{p,\varphi}(\Omega)}\leq C(\|f\|_{M_{p,\varphi}(\Omega)}+\|u\|_{M_{p,\varphi}(\Omega)}). \] The core of the matter is the study of continuity in the spaces \(M_{p,\varphi}\) of certain sublinear integrals and their commutators with BMO functions which, in turn, allows them to obtain continuity of Calderón--Zygmund operators in \(M_{p,\varphi}(\mathbb{R}^n)\). With these results, by using representation formulas of \(D_{ij}u\) in terms of such operators, and standard techniques from the theor…
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 6160854
zbMATH Keywords: generalized Morrey space
zbMATH Keywords: sublinear integral
zbMATH Keywords: Calderón-Zygmund integral
zbMATH Keywords: commutator
zbMATH Keywords: BMO
zbMATH Keywords: VMO
zbMATH Keywords: elliptic equation
zbMATH Keywords: Dirichlet problem
MaRDI profile type: Publication
full work available at URL: s11118-012-9299-4
OpenAlex ID: W2075870047
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DOI: 10.1007/S11118-012-9299-4
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description: scientific article; zbMATH DE number 6160854