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

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...

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Autores principales:	Georgi Boyadzhiev, Nikolai Kutev
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	strong maximum principle degenerate fully non-linear elliptic systems viscosity solutions Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/5f3fe1ff36cd4a928fa3f23558c219f4
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spelling	oai:doaj.org-article:5f3fe1ff36cd4a928fa3f23558c219f42021-11-25T18:17:48ZStrong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems10.3390/math92229852227-7390https://doaj.org/article/5f3fe1ff36cd4a928fa3f23558c219f42021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2985https://doaj.org/toc/2227-7390In 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> or generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula> solutions, while we prove it for semi-continuous ones.Georgi BoyadzhievNikolai KutevMDPI AGarticlestrong maximum principledegenerate fully non-linear elliptic systemsviscosity solutionsMathematicsQA1-939ENMathematics, Vol 9, Iss 2985, p 2985 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	strong maximum principle degenerate fully non-linear elliptic systems viscosity solutions Mathematics QA1-939
spellingShingle	strong maximum principle degenerate fully non-linear elliptic systems viscosity solutions Mathematics QA1-939 Georgi Boyadzhiev Nikolai Kutev Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
description	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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>2</mn></msup></semantics></math></inline-formula> or generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>1</mn></msup></semantics></math></inline-formula> solutions, while we prove it for semi-continuous ones.
format	article
author	Georgi Boyadzhiev Nikolai Kutev
author_facet	Georgi Boyadzhiev Nikolai Kutev
author_sort	Georgi Boyadzhiev
title	Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
title_short	Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
title_full	Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
title_fullStr	Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
title_full_unstemmed	Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems
title_sort	strong maximum principle for viscosity solutions of fully nonlinear cooperative elliptic systems
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/5f3fe1ff36cd4a928fa3f23558c219f4
work_keys_str_mv	AT georgiboyadzhiev strongmaximumprincipleforviscositysolutionsoffullynonlinearcooperativeellipticsystems AT nikolaikutev strongmaximumprincipleforviscositysolutionsoffullynonlinearcooperativeellipticsystems
_version_	1718411383770447872

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

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