The Dirichlet elliptic problem involving regional fractional Laplacian

Abstract

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^α_Ωu=f \qquad & {\rm in}\quad Ω,\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=g\quad & {\rm on}\quad \partial Ω, \end{array} \end{equation} where $\Omega$ is a bounded open domain in ${\mathbb{R}}^N$ ($N\ge 2$) with $C^2$ boundary $\partial \Omega$, $\alpha \in (\frac{1}{2},1)$ and the operator $(-\Delta {)}_{\Omega }^{\alpha }$ denotes the regional fractional Laplacian. We prove that when $g\equiv 0$, problem (\ref{0}) admits a unique weak solution in the cases that $f\in L^2(\Omega )$, $f\in L^1(\Omega ,{\rho }^{\beta }dx)$ and $f\in \mathcal{M}(\Omega ,{\rho }^{\beta })$, here $\rho (x)=\mathrm{dist}(x,\partial \Omega )$, $\beta =2\alpha -1$ and $\mathcal{M}(\Omega ,{\rho }^{\beta })$ is a space of all Radon measures $\nu$ satisfying ${\int }_{\Omega }{\rho }^{\beta }d|\nu |<+\infty .$ Finally, we provide an Integral by Parts Formula for the classical solution of (\ref{0}) with general boundary data $g$.

19 pages

Cited by in corpus (1)

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