Optimization problems in rearrangement classes for fractional $p$-Laplacian equations

TL;DR

This paper studies optimization problems involving the fractional p-Laplacian, proving existence of minimizers and analyzing maximization of energy functionals within rearrangement classes, with implications for nonlinear equations.

Contribution

It establishes the existence of minimizers for the principal eigenvalue and explores energy maximization in nonlinear fractional p-Laplacian problems within rearrangement classes.

Findings

Existence of minimizers for the principal eigenvalue.

Characterization of optimal data and solutions.

Analysis of energy maximization in nonlinear problems.

Abstract

We discuss two optimization problems related to the fractional $p$-Laplacian. First, we prove the existence of at least one minimizer for the principal eigenvalue of the fractional $p$-Laplacian with Dirichlet conditions, with a bounded weight function varying in a rearrangement class. Then, we investigate the maximization of the energy functional for general nonlinear equations driven by the same operator, as the reaction varies in a rearrangement class. In both cases, we provide a pointwise relation between the optimizing datum and the corresponding solution.