A one-phase problem for the fractional Laplacian: regularity of flat free boundaries

Daniela de Silva (1, 2) , Ovidiu Savin (2) , Yannick Sire (3)

1 Barnard College [Columbia University - New York]
2 Columbia University [New York]
3 I2M - Institut de Mathématiques de Marseille

We consider a one-phase free boundary problem involving a fractional Laplacian $(-\Delta)^\alpha$, $0<\alpha <1,$ and we prove that ''flat free boundaries" are $C^{1,\gamma}$. We thus extend the known result for the case $\alpha=1/2.$

Domaines: Equations aux dérivées partielles [math.AP]

Daniela de Silva, Ovidiu Savin, Yannick Sire. A one-phase problem for the fractional Laplacian: regularity of flat free boundaries. 2014. ⟨hal-01005328⟩