Liouville type theorems for nonlinear elliptic equations involving operator in divergence form

2012 ◽  
Vol 53 (10) ◽  
pp. 103706
Author(s):  
M. Hsini
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

2016 ◽  
Vol 8 (1) ◽  
pp. 193-202 ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Belgacem Rahal
Keyword(s):  
Elliptic Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlinear Elliptic Equations ◽  
Monotonicity Formula ◽  
Compact Set ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems ◽  
Nonexistence Of Solutions

Abstract In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Jiuyi Zhu
Keyword(s):  
Elliptic Equations ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Decay Estimates ◽  
Fully Nonlinear Elliptic Equations ◽  
Liouville Type ◽  
Fully Nonlinear Equations ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems

AbstractThe main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.

Elliptic equations on manifolds and isoperimetric inequalities

1990 ◽  
Vol 114 (3-4) ◽  
pp. 213-227 ◽  
Author(s):  
Andrea Cianchi
Keyword(s):  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
A Priori ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Isoperimetric Inequalities ◽  
Divergence Form ◽  
Sharp Estimates ◽  
Nonlinear Elliptic ◽  
Homogeneous Neumann Boundary Condition

SynopsisWe consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.

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