The Dirichlet elliptic problem involving regional fractional Laplacian

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad \Omega,\\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=g\quad & {\rm on}\quad \partial \Omega, \end{array} \end{equation} where $\Omega$ is a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$, $\alpha\in(\frac12,1)$ and the operator $(-\Delta)^\alpha_\Omega$ denotes the regional fractional Laplacian. We prove that when $g\equiv0$, 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 $\rho(x)={\rm dist}(x,\partial\Omega)$, $\beta=2\alpha-1$ 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 $g$.