The Dirichlet problem for a Petrovskiĭ-elliptic system of second-order equations
type: Item
instance of: scholarly article
title: The Dirichlet problem for a Petrovskiĭ-elliptic system of second-order equations
zbMATH Open document ID: 0939.35062
DOI: 10.1007/BF02674307
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: The apparatus of singular integral equations is applied to studying the Dirichlet problem for the system \[ -\Delta u_j + \lambda_j\frac{\partial}{\partial x_j}\sum_{i=1}^n \frac{\partial u_i}{\partial x_i} = 0,\qquad j=1,\dots, n. \] The main results of the article are as follows: Theorem 1. If the parameters \(\lambda_j\) of the system satisfy either the inequalities \(\lambda_j < 1\) or the inequalities \(\lambda_j >2\), then the Dirichlet problem for the system in an arbitrary half-space is solvable for arbitrary differentiable data and the solution is unique. Theorem 2. The Dirichlet problem for the system in an arbitrary convex domain with smooth boundary is of Fredholm type if either all \(\lambda_j<1\) or all \(\lambda_j>2\).
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 1432192
zbMATH Keywords: Petrovskij-elliptic system
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: method of singular integral equations
zbMATH Keywords: existence and uniqueness theorem
zbMATH Keywords: Fredholm type
author: A. I. Yanushauskas
MaRDI profile type: Publication
cites work: Q4003755
cites work: Q3843833
cites work: Pseudo-differential operators and non-elliptic boundary problems
cites work: Q3793075
cites work: Q5596448
description: scientific article; zbMATH DE number 1432192