Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations (original) (raw)

Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations

Potential Analysis

In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain Ω satisfies the exterior C 1,Dini condition at x 0 ∈ ∂Ω (see Definition 1.2), the solution is Lipschitz continuous at x 0 ; if Ω satisfies the interior C 1,Dini condition at x 0 (see Definition 1.4), the Hopf lemma holds at x 0. The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the C 1,Dini conditions are optimal.

Lipschitz regularity results for nonlinear strictly elliptic equations and applications

Journal of Differential Equations, 2017

Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.

Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms

arXiv (Cornell University), 2020

In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior and boundary pointwise C k,α regularity (k ≥ 1 and 0 < α < 1). In addition, we also derive the pointwise C k regularity (k ≥ 1) and C k,lnL regularity (k ≥ 0), which correspond to the end points α = 0 and α = 1 respectively. Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.

Comparison Principle for Elliptic Equations with Mixed Singular Nonlinearities

arXiv: Analysis of PDEs, 2019

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -\Delta_p u= \frac{f}{u^\gamma} + g u^q & \mbox{in Omega\OmegaOmega,} \\ u = 0 & \mbox{on partialOmega\partial\OmegapartialOmega,} \end{cases} \end{equation*} where Omega\OmegaOmega is an open bounded subset of mathbbRN\mathbb{R}^NmathbbRN, Deltapu:=textdiv(∣nablau∣p−2nablau)\Delta_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)Deltapu:=textdiv(nablaup2nablau) is the usual ppp-Laplacian operator, gammageq0\gamma\geq 0gammageq0 and 0leqqleqp−10\leq q\leq p-10leqqleqp1; fff and ggg are nonnegative functions belonging to suitable Lebesgue spaces.

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

Progress in Mathematics, 2020

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is C 1,1. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by C 1,1 solutions on larger and larger compact domains, and, in particular, for entire C 1,1 loc solutions: they are either constants or standard bubbles. Contents

Maximum principles for viscosity solutions of weakly elliptic equations

Bruno Pini Mathematical Analysis Seminar, 2019

Maximum principles play an important role in the theory of elliptic equations. In the last decades there have been many contributions related to the development of fully nonlinear equations and viscosity solutions. Here we consider degenerate elliptic equations, where the main term is a partial trace of the Hessian matrix of the solution. We establish maximum principles in domains that are unbounded in some directions, contained in slabs, and extended maximum principles, which lead to removable singularity results.

A strong comparison principle for positive solutions of degenerate elliptic equations

Differential and Integral Equations

A strong comparison principle (SCP, for brevity) is obtained for nonnegative weak solutions u ∈ W 1,p 0 (Ω) of the following class of quasilinear elliptic boundary value problems, (P) − div(a(x, ∇u)) − b(x, u) = f (x) in Ω; u = 0 on ∂Ω. Here, p ∈ (1, ∞) is a given number, Ω is a bounded domain in IR N with a connected C 2-boundary, a(x, ∇u) and b(x, u) are slightly more general than the functions a 0 (x)|∇u| p−2 ∇u and b 0 (x)|u| p−2 u, respectively, with a 0 ≥ const > 0 and b 0 ≥ 0 in L ∞ (Ω), and 0 ≤ f ∈ L ∞ (Ω). Validity of the SCP is investigated also in the case when b 0 ≤ 0 depending upon whether p ≤ 2 or p > 2. The methods of proofs are new.