Schwarz Symmetrization and Comparison Results for Nonlinear Elliptic Equations and Eigenvalue Problems

TL;DR

This paper investigates how Schwarz symmetrization compares solutions of nonlinear elliptic equations and eigenvalue problems in general domains with those in symmetric balls, providing bounds, existence results, and eigenvalue estimates.

Contribution

It introduces new comparison results using Schwarz symmetrization for nonlinear elliptic equations and eigenvalue problems, including bounds and existence criteria.

Findings

Solutions in general domains can be bounded by solutions in symmetric balls.

Existence and bounds of solutions are established for certain nonlinear equations.

A relationship between the first p-eigenvalue and the solution's supremum is demonstrated.

Abstract

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the existence and bound of solutions for some nonlinear equation. Moreover, for some nonlinear problems, we show that if the first $p$-eigenvalue of a domain is big, the supremum of a solution related to this domain is close to zero. For that we obtain $L^{\infty}$ estimates for solutions of nonlinear and eigenvalue problems in terms of other $L^p$ norms.