2019/06/14 by Lucia De Luca, De Luca, Lucia, Matteo Novaga +3 · 5 citations
Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #Functional Analysis (math.FA) #Mathematical functions and polynomials #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1906.06303
openalex publication_date 2019/06/14 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
This paper provides a unified point of view on fractional perimeters and\nRiesz potentials. Denoting by H^\σ - for \σ\∈ (0,1) - the\n\σ-fractional perimeter and by J^\σ - for \σ\∈ (-d,0) - the\n\σ-Riesz energies acting on characteristic functions, we prove that both\nfunctionals can be seen as limits of renormalized self-attractive energies as\nwell as limits of repulsive interactions between a set and its complement. We\nalso show that the functionals H^\σ and J^\σ ,, up to a suitable\nadditive renormalization diverging when \σ\→ 0, belong to a continuous\none-parameter family of functionals, which for \σ=0 gives back a new\nobject we refer to as it 0-fractional perimeter. All the convergence\nresults with respect to the parameter \σ and to the renormalization\nprocedures are obtained in the framework of \Γ-convergence. As a\nbyproduct of our analysis, we obtain the isoperimetric inequality for the\n0-fractional perimeter.\n