$s$-harmonic functions in the small order limit

Abstract

We study families $u_s$ of functions satisfying the equations $(-\hat{I}”{)}^s{u}_s=0$, $s\in (0,1)$ in a smooth bounded open set $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of the asymptotics of these families in the zero order limit $s\to 0^{+}$. Second, we study the differentiability of $u_s$ as a function of $s$. Most of our results are devoted to the associated Poisson problem, where the family $u_s$ is determined by the exterior condition $u_s=g$ in ${\mathbb{R}}^N\setminus \hat{I}\text{c◯}$ for some fixed function $g\in L^{\infty }({\mathbb{R}}^N\setminus \hat{I}\text{c◯})$. Our results show that both the zero order asymptotics and the differentiability properties of $u_s$ can be expressed in terms of the logarithmic Laplacian of suitable extensions of $g$. This allows to deduce pointwise monotonicity properties of $u_s$ in the order parameter $s$ for a large class of functions $g$.

40 pages, comments are welcome!