Boundary Singularities of Solutions to Semilinear Fractional Equations

We prove the existence of a solution of{(-\Delta)^{s}u+f(u)=0}in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functionsfsatisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where{f(u)=u^{p}}and μ is a Dirac mass, we show the existence of several critical exponentsp. We also demonstrate the existence of several types of separable solutions of the equation{(-\Delta)^{s}u+u^{p}=0}in{\mathbb{R}^{N}_{+}}.

Balls minimize trace constants in BV

Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincaré type boundary trace inequalities for functions of bounded variation with vanishing median or mean value.

Sharp boundary trace inequalities

This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ ℝN. The inequalities bound (semi-)norms of the boundary trace by certain norms of the function, its gradient on the region and by two specific constants κρ and κΩ associated with the domain and a weight function, respectively. These inequalities are sharp in that there exist functions for which equality holds. Explicit inequalities in some special cases when the region is a ball, or the region between two balls, are evaluated.