Free boundary problems

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Studied problems for $(-\Delta)^s$

Here are some free boundary problems for the fractional Laplacian operator $(-\Delta)^s$ that have been studied to some extent.

The extension technique,[1] representing the fractional Laplace operator as a suitable Dirichlet-to-Neumann operator on the boundary of the upper half-space in one more dimension, figures prominently in the study of many problems below.

Fractional obstacle problem

The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.

Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the limit of the Almgren frequency formula.[2][3]

Fractional Alt-Caffarelli problem

The problem is to seek the minimizer of an energy which is the sum of the Dirichlet form corresponding to the fractional Laplacian and the measure of the positivity set, taking the minimizer $u$ among nonnegative functions. It is usually formulated in terms of the extension.

Like its 2nd order analogue, solutions to problem are known to have optimal regularity of Holder class $C^s$, and to be $s$-harmonic away from the zero set. They have nondegeneracy of order $s$ as well, which is to say that $$ u(X) \geq c d(X,\Lambda)^s $$ where $\Lambda$ is the zero set of the $u$.

Unlike its 2nd order analogue, Hausdorff measure estimates of the free boundary are not known, but there are Lebesgue density estimates of the zero and free set in the neighborhood of a free boundary point. The free boundary is $C^{1,\alpha}$ about points where it is suitably flat.[4][5]