Minimization of Degenerate Nonlinear Functionals under Radial Symmetry

TL;DR

This paper proves the existence and radial symmetry of minimizers for nonlinear functionals with degenerate radial weights, extending previous methods to non-classical weight conditions in various dimensions.

Contribution

It introduces a new approach to establish minimizer existence and symmetry without relying on classical weight assumptions like doubling or Muckenhoupt conditions.

Findings

Existence of minimizers in a suitable functional class.

Minimizers exhibit radial symmetry.

Method applicable to weights with degenerate radial structure.

Abstract

In this work, we study the minimization of nonlinear functionals in dimension $d\geq 1$ that depend on a degenerate radial weight $w$. Our goal is to prove the existence of minimizers in a suitable functional class here introduced and to establish that the minimizers of such functionals, which exhibit $p$-growth with $1 < p < +\infty$, are radially symmetric. In our analysis, we adopt the approach developed in [Chiad\`o Piat, De Cicco and Melchor Hernandez, NoDEA $2025$, De Cicco and Serra Cassano, ESAIM:COCV $2024$], where $w$ does not satisfy classical assumptions such as doubling or Muckenhoupt conditions. The core of our method relies on proving the validity of a weighted Poincar\'e inequality involving a suitably constructed auxiliary weight.