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A non-local one-phase free boundary problem from obstacle to cavitation. (arXiv:1810.05535v1 [math.AP])
来源于:arXiv
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. 查看全文>>