Global gradient estimates for parabolic systems from composite materials
type: Item
instance of: scholarly article
title: Global gradient estimates for parabolic systems from composite materials
zbMATH Open document ID: 1401.35154
author: Yunsoo Jang
author: Youchan Kim
published in: Calculus of Variations and Partial Differential Equations
publication date: 2018-05-25T00:00:00Z
review text: The authors prove regularity estimates for weak solutions of parabolic systems with measurable coefficients in domains that are finite unions of subdomains with Reifenberg flat boundaries, precisely \(\Omega\) contains a finite union of subdomains \(\Omega_l, l = 1, \ldots, K\), whose boundaries are Reifenberg flat and \[ u_t^i - D_{\alpha}[A_{ij}^{\alpha \beta}D_{\beta} u^j ]= D_{\alpha}F^i_{\alpha}, \text{ in } \Omega_T = \Omega \times (0, T], \] \[ u^i = 0, \text{ on } \partial_p \Omega_T = \partial \Omega \times [0,t] \cup \Omega \times \{t = 0 \}. \] The coefficients are assumed to have BMO norm that is small in each subdomain, but can have large jumps on the boundaries. They remark that the estimates do not depend on the distance between the subdomains, and that the subdomains are allowed to touch.The results are proved for Orlicz norms that include results for \(L^p\). For example, if \(p \geq 2\), there is a \(\delta\), which satisfies \(0 < \delta < 1/8\), such that if the coefficients are \((\delta, R) \) composite vanishing and \(F\) is a vector field in \(L^p\), then \[ \int_0^T \int_{\Omega} |Du|^p \, dx dt \leq C \int_0^T \int_{\Omega} |F|^p \, dx dt. \] Here \((\delta, R)\) composite vanishing means that the BMO norm of the coefficients (now denoted \(A_{ij}^{\alpha \beta, l}\) on \(\Omega_l)\) is less than or equal to \(\delta\) over balls of radius less than \(R\) in each subdomain \(\Omega_l\). The BMO condition applies to each subdomain, but the BMO norm could be large if considered on \(\Omega\).
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6873459
zbMATH Keywords: smoothness and regularity of solutions
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: s00526-018-1330-1
OpenAlex ID: W2790653895
cites work: An Elliptic Regularity Result for a Composite Medium with "Touching" Fibers of Circular Cross-Section
cites work: Elliptic equations with measurable nonlinearities in nonsmooth domains
cites work: Parabolic Systems with Measurable Coefficients in Reifenberg Domains
cites work: Global gradient estimates in weighted Lebesgue spaces for parabolic operators
cites work: Parabolic systems with measurable coefficients in weighted Orlicz spaces
cites work: Elliptic equations with measurable coefficients in Reifenberg domains
cites work: Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains
cites work: Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients
cites work: Hessian estimates for fourth order elliptic systems with singular BMO coefficients in composite Reifenberg domains
cites work: Harmonic measure on locally flat domains
cites work: Global gradient estimates for parabolic equations with measurable nonlinearities
cites work: On the extension property of Reifenberg-flat domains
cites work: Estimates for elliptic systems from composite material
cites work: Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients
cites work: Elliptic equations with singular BMO coefficients in Reifenberg domains
cites work: Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains
cites work: Q4221203
cites work: Gradient estimates for \(p\)-Laplacian equation in composite reifenberg domains
DOI: 10.1007/S00526-018-1330-1
description: scientific article; zbMATH DE number 6873459