Higher regularity of the free boundary in the obstacle problem for the fractional heat operator. (English) Zbl 1532.35103
Summary: We prove a higher regularity result for the free boundary in the obstacle problem for the fractional heat operator via a higher order boundary Harnack estimate. As a consequence, we show that if the obstacle is \(H^{m + \beta}\), then the free boundary is \(H^{m-1 + \alpha}\) near non-degenerate free boundary points for some \(0 < \alpha \leq \beta\). In particular, smooth obstacles yield smooth free boundaries near non-degenerate free boundary points.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K85 | Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators |
| 35R11 | Fractional partial differential equations |
| 35R35 | Free boundary problems for PDEs |
| 47A57 | Linear operator methods in interpolation, moment and extension problems |
Keywords:
boundary Harnack estimate; extension problem; fractional heat operator; free boundary problemReferences:
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