A nonlocal free boundary problem

2014/11/28 by Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.7971

Abstract

Given~s,σ∈(0,1) and a bounded domain~Ω⊂\Rⁿ, we consider the following minimization problem of s-Dirichlet plus σ-perimeter type [u] _{H^s(\R²ⁿ∖(Ω^c)²)} + \Per_σ(\ugt;0\,Ω), where~[ ⋅]_H^s is the fractional Gagliardo seminorm and \Per_σ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper, as s\nearrow1, σ\nearrow1 or~σ\searrow0.

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