scholarly journals Comparison principle and Liouville type results for singular fully nonlinear operators

2004 ◽  
Vol 13 (2) ◽  
pp. 261-287 ◽  
Author(s):  
Isabeau Birindelli ◽  
Françoise Demengel
Keyword(s):  
Comparison Principle ◽  
Liouville Type ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Liouville Type Results

A Liouville-type theorem for fully nonlinear CR invariant equations on the Heisenberg group

2021 ◽  
pp. 2150060
Author(s):  
Bo Wang
Keyword(s):  
Heisenberg Group ◽  
Viscosity Solutions ◽  
Comparison Principle ◽  
Type Theorem ◽  
Liouville Type ◽  
Nonlinear Operators ◽  
Liouville Type Theorem ◽  
Fully Nonlinear

We obtain a Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group.

scholarly journals Ergodic pairs for singular or degenerate fully nonlinear operators

2019 ◽  
Vol 25 ◽  
pp. 75 ◽  
Author(s):  
Isabeau Birindelli ◽  
Françoise Demengel ◽  
Fabiana Leoni
Keyword(s):  
Comparison Principle ◽  
Order Term ◽  
A Priori ◽  
Zero Order ◽  
Dirichlet Problems ◽  
Zeroth Order ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Lipschitz Estimates ◽  
Ergodic Problem

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

Hadamard and Liouville Type Results for Fully Nonlinear Partial Differential Inequalities

2003 ◽  
Vol 05 (03) ◽  
pp. 435-448 ◽  
Author(s):  
I. Capuzzo Dolcetta ◽  
A. Cutrì
Keyword(s):  
Differential Inequalities ◽  
Liouville Type ◽  
Partial Differential ◽  
Fully Nonlinear ◽  
Whole Space ◽  
Liouville Type Results

In this paper we prove some Hadamard and Liouville type properties for nonnegative viscosity supersolutions of fully nonlinear uniformly elliptic partial differential inequalities in the whole space.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite

2017 ◽  
Vol 9 (4) ◽  
pp. 1
Author(s):  
Lina Wu
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Differential Forms ◽  
Computational Method ◽  
Liouville Type ◽  
Balanced Growth ◽  
Curved Manifolds ◽  
Research Findings ◽  
Liouville Type Results ◽  
Flat Manifolds

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincaré-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincaré-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincaré-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of $q$-energy from finite to infinite. At this point, a technique of $p$-balanced growth has been introduced to overcome difficulties of broadening from finite $q$-energy in $L^q$ spaces to infinite $q$-energy in non-$L^q$ spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincaré-Sobolev Inequality approaching to infinite $q$-energy growth.

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