Abnormal boundary decay for the fractional Laplacian
Soobin Cho, Renming Song
TL;DR
The paper identifies $C^{1,\text{Dini}}$ regularity as the sharp threshold for the standard boundary decay of nonnegative $\alpha$-harmonic functions and the Dirichlet heat kernel of the fractional Laplacian. It develops barrier-based methods and leverages Lévy system techniques to derive explicit decay rates governed by $\int_{\delta_D(x)}^R \frac{\ell(u)}{u}du$, connecting geometric regularity with quantitative boundary behavior. The authors construct the $D_{\ell}$ and $D_{-\ell}$ domains to show both upper and lower bounds and to exhibit counterexamples when the modulus is non-Dini, thereby proving optimality. The work extends boundary decay theory beyond $C^{1,\varepsilon}$ and $C^{1,1}$ domains, clarifying how subtle regularity in the boundary controls nonlocal decay rates with direct implications for potential theory and stochastic processes associated with $\Delta^{\alpha/2}|_D$. Overall, the results provide precise, sharp criteria for boundary phenomena in nonlocal elliptic and parabolic problems.
Abstract
In this paper, we show that, for $α\in (0,2)$, the $C^{1, \rm Dini}$ regularity assumption on an open set $D\subset \mathbb R^d$ is optimal for the standard boundary decay property for nonnegative $α$-harmonic functions in $D$ and for the standard boundary decay property of the heat kernel $p^D(t,x,y)$ of the Dirichlet fractional Laplacian $Δ^{α/2}|_D$ by proving the following: (i) If $D$ is a $C^{1, \rm Dini}$ open set and $h$ is a non-negative function which is $α$-harmonic in $D$ and vanishes near a portion of $\partial D$, then the rate at which $h(x)$ decays to 0 near that portion of $\partial D$ is ${\rm dist} (x, D^c)^{α/2}$. (ii) If $D$ is a $C^{1, \rm Dini}$ open set, then, as $x\to \partial D$, the rate at which $p^D(t,x,y)$ tends to 0 is ${\rm dist} (x, D^c)^{α/2}$. (iii) For any non-Dini modulus of continuity $\ell$, there exist non-$C^{1, \rm Dini}$ open sets $D$, with $\partial D$ locally being the graph of a $C^{1, \ell}$ function, such that the standard boundary decay properties above do not hold for $D$.