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

Abstract

We consider a one-phase free boundary problem of the minimizer of the energy \[ J_γ(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^γdx}, \] 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.

arXiv admin note: text overlap with arXiv:1401.6443, arXiv:1102.3086 by other authors