Non-local, non-convex functionals converging to Sobolev norms

2019/09/05 by Brezis, Haim, Nguyen, Hoai-Minh
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1909.02160

Abstract

We study the pointwise convergence and the Γ-convergence of a family of non-local, non-convex functionals Λ_δ in L^p(Ω) for p>1. We show that the limits are multiples of ∫_Ω |∇ u|^p. This is a continuation of our previous work where the case p=1 was considered.

Related

• Γ-convergence of non-local, non-convex functionals in one dimension
• Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms
• Continuity of some non-local functionals with respect to a convergence of the underlying measures
• On the shape factor of interaction laws for a non-local approximation of\n the Sobolev norm and the total variation
• On some non-local approximation of nonisotropic Griffith-type functionals