Search results for "Elliptic equations"

A weak comparison principle for solutions of very degenerate elliptic equations

2012

We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where \(\nabla u\in K\), where \(K\subset \mathbb{R }^N\) is a Borel set containing the origin.

Bounded solutions to the 1-Laplacian equation with a critical gradient term

2012

Uniqueness of positive radial solutions to singular critical growth quasilinear elliptic equations

2015

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle \frac{\mu}{|x|^{p}}|u|^{p-2}u}{\displaystyle =\frac{|u|^{\frac{(N-s)p}{N-p}-2}u}{|x|^{s}}}+\lambda|u|^{p-2}u & \text{in }B,\\ u=0 & \text{on }\partial B, \end{cases} \] where $B$ is an open finite ball in $\mathbb{R}^{N}$ centered at the origin, $1<p<N$, $-\infty<\mu<((N-p)/p)^{p}$, $0\le s<p$ and $\lambda\in\mathbb{R}$. A related limiting problem is also considered.

Improved Hölder regularity for strongly elliptic PDEs

2019

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of H\"older regularity, higher than what is given by the classical exponent $1/K$.

Discontinuous solutions of linear, degenerate elliptic equations

2008

Abstract We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.

A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term

2017

This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary H\"older continuity and a boundary Harnack type inequality.

Scheduled Relaxation Jacobi method: improvements and applications

2016

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…

Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

2022

Abstract We consider positive critical points of Caffarelli–Kohn–Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted p -Laplace operator, which we consider for a general p ∈ ( 1 , d ) . For p = 2 , the symmetry breaking region for extremals of Caffarelli–Kohn–Nirenberg inequalities was completely characterized in Dolbeault et al. (2016). Our results extend this result to a general p and are optimal in some cases.

Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential

2015

Abstract Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations − Δ p u − μ | x | p | u | p − 2 u = Q ( x ) | u | N p N − p − 2 u , x ∈ R N , where 1 p N , 0 ≤ μ ( ( N − p ) / p ) p and Q ∈ L ∞ ( R N ) .

Generalized Harnack inequality for semilinear elliptic equations

2015

Abstract This paper is concerned with semilinear equations in divergence form div ( A ( x ) D u ) = f ( u ) , where f : R → [ 0 , ∞ ) is nondecreasing. We introduce a sharp Harnack type inequality for nonnegative solutions which is a quantified version of the condition for strong maximum principle found by Vazquez and Pucci–Serrin in [30] , [24] and is closely related to the classical Keller–Osserman condition [15] , [22] for the existence of entire solutions.