Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

被引:1
作者
Boyadzhiev, Georgi [1 ,2 ]
Kutev, Nikolai [1 ]
机构
[1] Inst Math & Informat, 8 Acad Georgi Bonchev Str, Sofia 1113, Bulgaria
[2] Univ Architecture Civil Engn & Geodesy, Sofia 1046, Bulgaria
关键词
strong maximum principle; degenerate fully non-linear elliptic systems; viscosity solutions;
D O I
10.3390/math9222985
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C(2 )or generalized C(1 )solutions, while we prove it for semi-continuous ones.
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页数:9
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共 23 条
[1]  
Andersson L, 1998, COMMUN PUR APPL MATH, V51, P581, DOI 10.1002/(SICI)1097-0312(199806)51:6<581::AID-CPA2>3.3.CO
[2]  
2-E
[3]   DIFFERENTIAL-GAMES WITH MAXIMUM COST [J].
BARRON, EN .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1990, 14 (11) :971-989
[4]   THE BELLMAN EQUATION FOR MINIMIZING THE MAXIMUM COST [J].
BARRON, EN ;
ISHII, H .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1989, 13 (09) :1067-1090
[5]   THE PRINCIPAL EIGENVALUE AND MAXIMUM PRINCIPLE FOR 2ND-ORDER ELLIPTIC-OPERATORS IN GENERAL DOMAINS [J].
BERESTYCKI, H ;
NIRENBERG, L ;
VARADHAN, SRS .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1994, 47 (01) :47-92
[6]   STRONG INTERIOR AND BOUNDARY MAXIMUM PRINCIPLE FOR WEAKLY COUPLED LINEAR COOPERATIVE ELLIPTIC SYSTEMS [J].
Boyadzhiev, Georgi ;
Kutev, Nikolay .
COMPTES RENDUS DE L ACADEMIE BULGARE DES SCIENCES, 2019, 72 (07) :861-870
[7]   Strong maximum principle for nonlinear cooperative elliptic systems [J].
Boyadzhiev, Georgi ;
Kutev, Nikolay .
SIXTH INTERNATIONAL CONFERENCE NEW TRENDS IN THE APPLICATIONS OF DIFFERENTIAL EQUATIONS IN SCIENCES (NTADES 2019), 2019, 2159
[8]   On maximum principles for cooperative elliptic systems via fixed point index [J].
Correa, FJSA ;
Souto, MAS .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1996, 26 (05) :997-1006
[9]   USERS GUIDE TO VISCOSITY SOLUTIONS OF 2ND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS [J].
CRANDALL, MG ;
ISHII, H ;
LIONS, PL .
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1992, 27 (01) :1-67
[10]  
De Figueiredo D., 2013, SELECTED PAPERS, P36