Old and new results in regularity theory for diagonal elliptic systems via blowup techniques
type: Item
instance of: scholarly article
title: Old and new results in regularity theory for diagonal elliptic systems via blowup techniques
zbMATH Open document ID: 1329.35145
author: Lisa Beck
author: Miroslav Bulíček
author: Jens Frehse
published in: Journal of Differential Equations
publication date: 2015-09-22T00:00:00Z
review text: The authors consider the diagonal elliptic system \[ \begin{alignedat}{2} & -\sum_{i,j=1}^d D_i(a_{ij}(x) D_j u_{\nu}(x)) + \kappa_{\nu} u_{\nu}(x) = H_{\nu}(x,u(x), \nabla u(x)) \qquad & \text{in } & \Omega,\\ & u_{\nu}(x) = 0 \qquad & \text{on } & \Gamma_D,\\ & \sum_{i,j=1}^d a_{ij}(x) D_j u_{\nu}(x) n_i(x) \qquad & \text{on } & \Gamma_N\end{alignedat} \] for \(\nu = 1, \ldots, N\) subject to mixed boundary conditions. Here \(\Gamma_{D}\) denotes the Dirichlet and \(\Gamma_{N}\) the Neumann part of the boundary. Further, \(\Omega \subset \mathbb{R}^d\) for \(d \geq 2\) is assumed to be a bounded Lipschitz domain, \(\kappa \in \mathbb{R}^N\) is a fixed vector with positive components and \(H: \Omega \times \mathbb{R}^N \times \mathbb{R}^{d \times N} \to \mathbb{R}^N\) is a nonlinearity that couples the equations. The system is assumed to be diagonal, i.e.\ \(a: \Omega \to \mathbb{R}^{d \times d}\), with bounded and measurable components. The equation is studied for nonlinearities \(H\) with critical quadratic growth in the gradient variable for which the existence of weak solutions may fail. It is shown that if \(H\) additionally satisfies estimates of the form \[ H_{\nu}(x,u,z) \leq K^* + K |z_{\nu}| |z| \] \[ \sum_{\nu = 1}^N q_{\nu} H_{\nu}(x,u,z) \geq -K^* - K \bigg| \sum_{\nu = 1}^{N} q_{\nu} z_{\nu} \bigg|, \] then the equation has a unique weak solution (which is defined in a slightly more general sense than usual) that is Hölder continuous for some fixed constant on open subsets of \(\Omega\) with positive distance to the Dirichlet part of the boundary. Further, \(W^{2,q}(\Omega)\)-estimates are obtained for \(C^{1,1}\) domains and coefficients \(a_{ij} \in C^{0,1}(\overline{\Omega})\). As key steps a Liouville theorem for solutions and a compactness result for weak solutions are obtained via blow-up techniques.
reviewed by: Stephan Fackler
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 6484247
zbMATH Keywords: elliptic diagonal systems
zbMATH Keywords: existence
zbMATH Keywords: regularity
zbMATH Keywords: Hölder continuity
zbMATH Keywords: Liouville theorem
zbMATH Keywords: differential game theory
zbMATH Keywords: Dirichlet boundary
zbMATH Keywords: Neumann boundary
zbMATH Keywords: mixed boundary conditions
MaRDI profile type: Publication
full work available at URL: j.jde.2015.07.030
OpenAlex ID: W1473004709
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DOI: 10.1016/J.JDE.2015.07.030
description: scientific article; zbMATH DE number 6484247