# $C^{\alpha}$ estimates for spectral fractional Laplacian

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 \ \ \left\{\begin{aligned} (-\Delta)^su_{\epsilon}&=f_{\epsilon} &&\text{in } \Omega_{\epsilon} \\ u_{\epsilon} & =g_{\epsilon} &&\text{ in } \partial \Omega_{\epsilon} \end{aligned} \right. 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.

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 non-local 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 non-zero boundary condition $$g_\epsilon$$? (In the above I assumed that $$g_\epsilon = 0$$).
• My $g_{\epsilon}$ are non-zero. I am not sure if the problem is well-posed. One way might be to decompose the problem in to two parts. One with zero boundary condition and the others the s-harmonic part with non-zero boundary conditions. – Spal Jun 25 at 20:44
• @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. – Mateusz Kwaśnicki Jun 27 at 11:00
• 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. – Spal Jun 27 at 19:58