Minimization of Degenerate Nonlinear Functionals under Radial Symmetry

2025/07/28 by Piat, Valeria Chiadò, De Cicco, Virginia, Hernandez, Anderson Melchor
#26A15 #49J45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2507.20603

Abstract

In this work, we study the minimization of nonlinear functionals in dimension d≥ 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 < +∞, are radially symmetric. In our analysis, we adopt the approach developed in [Chiadò 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 Poincaré inequality involving a suitably constructed auxiliary weight.

Citations

• Relaxation for a degenerate functional with linear growth in the onedimensional case
• Metric Sobolev spaces I: equivalence of definitions
• Relaxation for degenerate nonlinear functionals in the onedimensional case
• Weighted Sobolev Spaces on Metric Measure Spaces
• Sobolev spaces in metric measure spaces: reflexivity and lower\n semicontinuity of slope
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Perimeter of sublevel sets in infinite dimensional spaces

Related

• Relaxation for degenerate nonlinear functionals in the onedimensional case
• On Symmetry of Minimizers in Constrained Quasi-Linear Problems, 1
• On the symmetry of minimizers in constrained quasi-linear problems
• On a class of variational problems with linear growth and radial\n symmetry
• Radial symmetry of minimax critical points for nonsmooth functionals