Work

$$\Gamma $$-Limsup estimate for a nonlocal approximation of the Willmore functional

Year: 2025

Type: article

Abstract: We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation of the fractional Allen–Cahn energies, and we prove the corresponding $$\Gamma $$ Γ -limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen–Cahn energies.

Source: Calculus of Variations and Partial Differential Equations

DOI: 10.1007/s00526-025-03039-w

Authors: Hardy Chan, Mattia Freguglia, Marco Inversi

Institutions: University of Basel, Scuola Normale Superiore

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190