A symmetry result for polyharmonic problems with Navier conditions

TL;DR

This paper proves that solutions to certain polyharmonic problems in symmetric, convex domains are themselves symmetric and monotone, extending symmetry results to higher-order elliptic equations with Navier boundary conditions.

Contribution

It establishes a symmetry and monotonicity result for polyharmonic problems with Navier conditions in domains that are convex and symmetric, generalizing known results to higher-order cases.

Findings

Solutions are symmetric under domain reflection.

Solutions are monotone in the symmetric direction.

Symmetry holds for any order of polyharmonic problems.

Abstract

We consider an elliptic polyharmonic problem of any order which takes place in a punctured bounded domain with Navier conditions. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperpane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction.