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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.

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(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 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 $g_\epsilon = 0$).

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  • $\begingroup$ 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. $\endgroup$ – Spal Jun 25 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. $\endgroup$ – Mateusz Kwaśnicki Jun 27 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. $\endgroup$ – Spal Jun 27 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; (...) $\endgroup$ – Mateusz Kwaśnicki Jun 27 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. $\endgroup$ – Mateusz Kwaśnicki Jun 27 at 20:12

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