Consider the following problem with $\Omega_{\epsilon}$ is bounded for each fixed $\epsilon>0$ and $s\in (0, 1)$. Let $u_{\epsilon}$ be a classical solution of \begin{equation} \ \ \left\{\begin{aligned} (\Delta)^su_{\epsilon}&=f_{\epsilon} &&\text{in } \Omega_{\epsilon} \\ u_{\epsilon} & =g_{\epsilon} &&\text{ in } \partial \Omega_{\epsilon} \end{aligned} \right. \end{equation} Suppose $\Omega_{\epsilon}\to \mathbb R^N$ as ${\epsilon}\to 0.$ Let $u_{\epsilon}, f_{\epsilon}$ is uniformly bounded in every compact set. Is it possible to derive local uniform $C^{\alpha}$ estimates for $u_{\epsilon}$. Any reference is welcome.
1 Answer
(Too long for a comment.)
Not sure if this will work for you: you can "cheat" and switch to the fractional Laplacian in full space. I mean, if you extend your $u_\epsilon$ to all of $\mathbb{R}^N$ so that $u_\epsilon(x) = 0$ for $x \notin \Omega_\epsilon$, then $$(\Delta)^s u_\epsilon = \tilde f_\epsilon$$ for some $\tilde f_\epsilon$, which is not much different from $f_\epsilon$ except near the boundary of $\Omega_\epsilon$. I mean, $$\tilde f_\epsilon = f_\epsilon + ((\Delta)^s  (\Delta_{\Omega_\epsilon})^s) u_\epsilon,$$ and the kernel of the nonlocal operator $(\Delta)^s  (\Delta_{\Omega_\epsilon})^s$ is given by $$K(x, y) = c_{N,s} \int_0^\infty (p_t(y  x)  p_t^{\Omega_\epsilon}(y  x)) t^{1  s} ds ,$$ where $p_t$ is the heat kernel for $\Delta$, while $p_t^\Omega$ is the Dirichlet heat kernel for $\Delta$ in $\Omega$. A simple probabilistic bound for $p_t(y  x)  p_t^\Omega(x, y)$ in terms of the hitting time of $\partial \Omega$ should now give a decent estimate for $K(x, y)$, and the result you need should follow from standard regularity theory for $(\Delta)^s$. This ideas are inspired by a series of articles by Song and Vondraček, see, e.g., here.
By the way, how exactly is your "spectral fractional Laplacian" defined so that it includes nonzero boundary condition $g_\epsilon$? (In the above I assumed that $g_\epsilon = 0$).

$\begingroup$ My $g_{\epsilon}$ are nonzero. I am not sure if the problem is wellposed. One way might be to decompose the problem in to two parts. One with zero boundary condition and the others the sharmonic part with nonzero boundary conditions. $\endgroup$– SpalJun 25, 2019 at 20:44

$\begingroup$ @Spal: From my point of view, a natural way would be to consider the "regional fractional Laplacian", which corresponds to the quadratic form $c_{n,s} \int_\Omega \int_\Omega (u(x)  u(y))^2 x  y^{n  2s} dx dy$. However, the choice of the operator really depends on what one is trying to describe. $\endgroup$ Jun 27, 2019 at 11:00

$\begingroup$ What is the difference between a spectral fractional Laplacian and a regional fractional Laplacian? I know when $\Omega=\mathbb R^N$ then the definition coincide. How do the two Laplacians behave locally. I believe they should behave in the same way, for example the Harnack inequality, local Schauder kind of estimates. $\endgroup$– SpalJun 27, 2019 at 19:58

$\begingroup$ @Spal: They are similar in the bulk, but much different near the boundary. There is a number of papers that try to discuss the differences; off the top of my head, I remember: (1) S. Duo and H. Wang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst., Ser. B, 24(1) (2019), 231–256; (2) P. Garbaczewski and V. A. Stephanovich, Fractional Laplacians in bounded domains: killed, reflected, censored and taboo Lévy flights, Preprint, 2018, arXiv:1810.07028; (...) $\endgroup$ Jun 27, 2019 at 20:11

$\begingroup$ (3) A. Lischke et al., What is the fractional Laplacian?, Preprint, 2018, arXiv:1801.09767; (4) my survey 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. $\endgroup$ Jun 27, 2019 at 20:12