2
$\begingroup$

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 $0<s<1.$

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.