Ergodic pairs for singular or degenerate fully nonlinear operators

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.

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Comparison principle and Liouville type results for singular fully nonlinear operators

Annales de la faculté des sciences de Toulouse Mathématiques ◽

10.5802/afst.1070 ◽

2004 ◽

Vol 13 (2) ◽

pp. 261-287 ◽

Cited By ~ 37

Author(s):

Isabeau Birindelli ◽

Françoise Demengel

Keyword(s):

Comparison Principle ◽

Liouville Type ◽

Nonlinear Operators ◽

Fully Nonlinear ◽

Liouville Type Results

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199721500607 ◽

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.

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A priori estimates for elliptic equations with gradient dependent term and zero order term

Journal of Differential Equations ◽

10.1016/j.jde.2021.08.037 ◽

2021 ◽

Vol 302 ◽

pp. 550-584

Author(s):

A. Alvino ◽

M.F. Betta ◽

A. Mercaldo ◽

R. Volpicelli

Keyword(s):

Elliptic Equations ◽

A Priori Estimates ◽

Order Term ◽

A Priori ◽

Zero Order ◽

Dependent Term ◽

Priori Estimates

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ZERO ORDER PERTURBATIONS TO FULLY NONLINEAR EQUATIONS: COMPARISON, EXISTENCE AND UNIQUENESS

Communications in Contemporary Mathematics ◽

10.1142/s0219199709003296 ◽

2009 ◽

Vol 11 (01) ◽

pp. 131-164 ◽

Cited By ~ 8

Author(s):

FERNANDO CHARRO ◽

IRENEO PERAL

Keyword(s):

Nonlinear Equations ◽

Comparison Principle ◽

Existence And Uniqueness ◽

Existence Of Solutions ◽

Elliptic Operators ◽

Zero Order ◽

Infinity Laplacian ◽

Fully Nonlinear Equations ◽

Fully Nonlinear ◽

First Case

We study existence of solutions to [Formula: see text] where F is elliptic and homogeneous of degree m, and either f(λ,u) = λ uqor f(λ,u) = λ uq+ ur, for 0 < q < m < r, and λ > 0. Furthermore, in the first case, we obtain that the solution is unique as a consequence of a comparison principle up to the boundary. Several examples, including uniformly elliptic operators and the infinity-laplacian, are considered.

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Regularizing effect of the interplay between coefficients in Dirichlet problems with convection or drift terms

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021015 ◽

2021 ◽

Author(s):

Lucio Boccardo

Keyword(s):

Diffusion Equation ◽

Order Term ◽

Zero Order ◽

Dirichlet Problems ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

The Subject ◽

Drift Terms

There are very important results by Enrique Zuazua on the subject of the convection-diffusion equation. In some sense this paper deals with a linear elliptic counterpart of the above equation if $d$ is not constant. We prove regularizing results on the solutions, under assumptions of interplay between the datum and the coefficient of the zero order term or between the modulus of the drift and the coefficient of the zero order term.

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

Journal of Geometric Analysis ◽

10.1007/s12220-021-00607-2 ◽

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.

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An asymptotic treatment for non-convex fully nonlinear elliptic equations: Global Sobolev and BMO type estimates

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500530 ◽

2019 ◽

Vol 21 (07) ◽

pp. 1850053 ◽

Cited By ~ 1

Author(s):

J. V. da Silva ◽

G. C. Ricarte

Keyword(s):

Elliptic Equations ◽

A Priori Estimates ◽

Integrability Condition ◽

A Priori ◽

Nonlinear Elliptic Equations ◽

Borderline Case ◽

Fully Nonlinear Elliptic Equations ◽

Fully Nonlinear ◽

Text Type ◽

Nonlinear Elliptic

In this paper, we establish global Sobolev a priori estimates for [Formula: see text]-viscosity solutions of fully nonlinear elliptic equations as follows: [Formula: see text] by considering minimal integrability condition on the data, i.e. [Formula: see text] for [Formula: see text] and a regular domain [Formula: see text], and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting “fine” regularity estimates from a limiting operator, the Recession profile, associated to [Formula: see text] to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when [Formula: see text]. In such a scenery, we show that solutions admit [Formula: see text] type estimates for their second derivatives.

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Gradient estimates for Monge–Ampère type equations on compact almost Hermitian manifolds with boundary

Analysis ◽

10.1515/anly-2021-0047 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Masaya Kawamura

Keyword(s):

Nonlinear Equations ◽

Smooth Solution ◽

A Priori ◽

Gradient Estimates ◽

Manifolds With Boundary ◽

Fully Nonlinear Equations ◽

Fully Nonlinear ◽

Hermitian Manifolds ◽

Almost Hermitian Manifolds ◽

Monge Ampere

Abstract We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

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