Preprint

Inserted: 28 jul 2025
Last Updated: 28 jul 2025

Year: 2025

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 (De Cicco, Serra Cassano), (Chiadò Piat, De Cicco, Melchor), 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 Poincaré inequality involving a suitably constructed auxiliary weight.

Keywords: Lower semicontinuity, degenerate variational integrals, Poincaré inequality

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