A non-local one-phase free boundary problem from obstacle to cavitation

TL;DR

This paper studies a fractional free boundary problem that interpolates between obstacle and cavitation problems, establishing homogeneity of blow-ups and regularity of the free boundary.

Contribution

It introduces a new fractional free boundary model and proves homogeneity of blow-ups and regularity results for the free boundary.

Findings

Blow-up near free boundary points is homogeneous of degree 2s/(2-γ).

Flat free boundary is C^{1,θ} when γ is close to 0.

The problem bridges obstacle and cavitation problems in fractional settings.

Abstract

We consider a one-phase free boundary problem of the minimizer of the energy \[ J_{\gamma}(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^{\gamma}dx}, \] with constants $0<s,\gamma<1$. It is an intermediate case of the fractional cavitation problem (as $\gamma=0$) and the fractional obstacle problem (as $\gamma=1$). We prove that the blow-up near every free boundary point is homogeneous of degree $\beta=\frac{2s}{2-\gamma}$, and flat free boundary is $C^{1,\theta}$ when $\gamma$ is close to 0.