Symmetrization with respect to the anisotropic perimeter and applications

TL;DR

This paper introduces a novel symmetrization method that preserves anisotropic perimeter, enabling sharp comparison results for solutions of certain nonlinear elliptic PDEs and anisotropic Hessian integrals.

Contribution

It presents a new symmetrization technique based on anisotropic perimeter preservation, advancing comparison principles in nonlinear elliptic PDE analysis.

Findings

Established sharp comparison results for fully nonlinear elliptic equations.

Developed a symmetrization method that preserves anisotropic perimeter.

Applied the method to anisotropic Hessian integrals.

Abstract

In this paper we introduce a new type of symmetrization, which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function, in order to prove sharp comparison results for solutions of a class of homogeneous Dirichlet fully nonlinear elliptic problems of second order and for suitable anisotropic Hessian integrals.