Title: The obstacle problem for the fractional Laplacian with drift
Abstract: We present the $C^{1,\alpha}$ regularity of the
regular part of the free boundary
in the obstacle problem defined by the fractional Laplacian
operator with gradient perturbation, in the subcritical regime
$(s\in (1/2,1))$. More specifically, we consider
\[
\min\{Lu,u- \varphi \}=0,
\]
where we denote
$Lu:=(-\Delta)^su+\langle b(x),\nabla u\rangle+c(x)u$.
Our proof relies on a new Weiss-type monotonicity formula and an
epiperimetric inequality. Both are generalizations of the ideas of
G. Weiss, used in the classical obstacle problem for the Laplace operator,
to our framework of fractional powers of the Laplace operator with drift.
This is joint work with Nicola Garofalo, Arshak Petrosyan and Camelia Pop