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Here are some free boundary problems for the [[fractional Laplacian]] operator $(-\Delta)^s$ that have been studied to some extent. The [[extension technique]],<ref name="CS"/> representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundar ...

...is construction is frequently used to turn nonlocal problems involving the fractional Laplacian into local problems in one more space dimension. ==Fractional Laplacian== ...

=== Traveling fronts in Fisher-KPP equations with fractional diffusion have exponential speed === ...ence can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate <ref name="CR"/><r ...

* Fractional curvatures in conformal geometry. ...d to explain further the [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geom ...

Consider two continuous functions $u$ and $v$ such that ...and $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that th ...

A similar argument works for the fractional Laplacian to prove $u \in C^{2s}$. However, this regularity is not optimal ...be achieved constructing appropriate barriers. The Poisson kernel of the [[fractional Laplacian]] is explicit in the upper half plane, which allows us to make di ...

on the closure of all $L^2$-functions with respect to $J(u)=E(u,u)$. Note characteristic functions of a set of positive measure do not belong to $H^1$. ...

...means in an integro-differential equation. The prime example will be the [[fractional Laplacian]]. For $s \in (0,2)$, define ...perators]] would be to impose that the kernel is comparable to that of the fractional Laplacian. The condition could be ...