Let $u$ be a solution of $-\Delta u= f \text{ in } \Omega$ with $u=0 \text{ on } \partial \Omega.$ If $f\in L^2(\Omega)$, then $u\in H^2(\Omega)\cap H^{1}_{0}(\Omega)$provided $\Omega$ is a smooth bounded domain.

Does this type of theorem holds true for the equation $(-\Delta)^s u= f \text{ in } \Omega$ with $u=0 \text{ in } \mathbb R^N -\Omega$ with $0<s<1.$


In your case, $u$ is locally in the fractional Sobolev space $H^{2s}$ (inside $\Omega$), and globally in $H^s$. The latter is either an assumption (in the weak formulation of the problem) or a proposition (when a different notion of a solution is used). The former follows already from the result given in Stein's book:

E. M. Stein, Singular Integrals and Differentiability Properties Of Functions, Princeton University Press, Princeton, 1970.

Similar questions were studied in much more detail by Xavier Ros-Oton and Joaquim Serra (with smoother $f$, though) in their paper:

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014): 275–302.

You may also have a look at the survey paper:

X. Ros-Oton, Nonlocal equations in bounded domains: a survey, Publ. Mat. 60(1) (2016): 3–26.

The first paper which deals with boundary regularity of $u$ is likely due to Krzysztof Bogdan:

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Stud. Math. 123(1) (1997): 43–80.

You can also find these references in my survey:

M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019.

  • $\begingroup$ What should be the norm on $H^{2s} (\Omega) \cap H^{s}_{0} (\Omega)$ if $s\in (0, 1).$ I believe it should be the sum of the $L^2(\Omega)$ norm of $(-\Delta)^s u$, $(-\Delta)^{s/2} u$ and $u.$ $\endgroup$
    – GabS
    May 10 '19 at 13:39
  • $\begingroup$ @GabS: Not sure what is the question (I mean, what kind of norm are you looking for?), but I do not think $u$ is in $H^{2s}(\Omega)$ near the boundary. In 1-D the function $u(x) = (1 - x^2)^s$ in $\Omega = (-1, 1)$ is the solution of $(-\Delta)^s u = c$ for an appropriate constant $c$, but unless I made a mistake, it is easy to check that $u$ is not in $H^{2s}(\Omega)$. $\endgroup$ May 10 '19 at 17:36
  • $\begingroup$ Consider the problem in the bounded domain $(-\Delta)^s u=f$ in $\Omega$ and $u=0$ in the complement of $\Omega$. If $f\in L^2(\Omega),$ is it true that $\|u\|_{H^{2s}(\Omega)\cap H^{s}_{0}(\Omega)}\leq C \|f\|_{L^2(\Omega)}.$ $\endgroup$
    – GabS
    May 10 '19 at 18:14
  • $\begingroup$ @GabS I think the example that I gave in my previous comment shows that $u$ need not have finite $H^{2s}(\Omega)$ norm. $\endgroup$ May 10 '19 at 19:01

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