Criteria for validity of the maximum modulus principle for solutions of linear strongly elliptic second order systems
type: Item
instance of: scholarly article
title: Criteria for validity of the maximum modulus principle for solutions of linear strongly elliptic second order systems
zbMATH Open document ID: 0820.35017
DOI: 10.1007/BF01047674
published in: Potential Analysis
publication date: 1993-06-29T00:00:00Z
review text: The systems under consideration are of the form (E) \(\sum A_{ij} \partial^ 2_{ij} u - \sum A_ j \partial_ ju - A_ 0u = 0\), where the \(A\)'s are sufficiently smooth \(m \times m\) matrix valued functions in \(\Omega \subset \mathbb{R}^ n\). The maximum modulus principle can be stated in two ways: 1) \(\| u \|_{C (\omega)} \leq \| u \|_{C (\partial \omega)}\) for all solutions of (E) in \(\Omega\) and all sufficiently regular \(\omega \subset \Omega\). 2) (E) has no solution with modulus attaining its strict local maximum in the interior of \(\Omega\). Either formulation is equivalent to the following condition: \(A_{ij} = Aa_{ij}\), where \(A\) is an \(m \times m\) matrix, \(a_{ij}\) are scalars and \(\sum a_{jk} \xi_ j \cdot \xi_ k + \sum A^{-1} A_ j \xi_ j \cdot \zeta + A^{-1} A_ 0 \zeta \cdot \zeta \geq 0\) for all \(\xi_ j\), \(\zeta \in \mathbb{R}^ m\) such that \(\xi_ j \cdot \zeta = 0\). It is shown that the condition 1) for \(\omega = \Omega\) alone does not imply the asserted form of the matrices \(A_{ij}\). The above result is a consequence and extension of the authors' previous work for systems with constant coefficients [Math. USSR, Sb. 53, 457-479 (1986); transl. from Mat. Sb., Nov. Ser. 125, No. 4, 458-480 (1984; Zbl 0577.35003)].
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 220431
zbMATH Keywords: maximum modulus principle
author: Gershon Kresin
author: V. Maz'ya
MaRDI profile type: Publication
cites work: Sui sistemi di equazioni lineari a derivate parziali del secondo ordine dei tipi ellittico e parabolico
cites work: Q3244626
cites work: Q3263019
cites work: Q5617915
cites work: Maximum principles for a class of first-order elliptical systems
cites work: Q3820984
cites work: Q3699208
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
cites work: Q3843833
full work available at URL: https://doi.org/10.1007/bf01047674
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description: scientific article; zbMATH DE number 220431