A construction of two different solutions to an elliptic system

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:465

Article

Cyranka, Jacek^1,2  Mucha, Piotr Boguslaw¹ 

[1] Univ Warsaw, Inst Appl Math & Mech, Banacha 2, PL-02097 Warsaw, Poland

[2] Rutgers State Univ, Dept Math, 110 Frelinghusen Rd, Piscataway, NJ 08854 USA

关键词: Nonlinear elliptic problem;    2D stationary Burgers equation;    Nonuniqucness;    Construction of two solutions;    Large matrices;   

DOI  :  10.1016/j.jmaa.2018.05.010

来源: Elsevier

【 摘 要 】

The paper aims at constructing two different solutions to an elliptic system u . del u + (-Delta)(m)u = lambda F defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual properties of the linear operator lambda sin y partial derivative(x)w + (-Delta)(m)w arising from a linearization of the equation about the dominant part of F. We argue that the skew-symmetric part of the operator provides in some sense a smallness of norms of the linear operator inverse. Our analytical proof is valid for a particular force F and for lambda > lambda(0), m > m(0) sufficiently large. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. Our analytical results are illustrated by numerical simulations. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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