局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性

In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^αΩ on a bounded open set Ω ■R(N ≥ 2) with C^(1,1) boundary ■Ω. We prove that when f ∈ L^p(Ω), and g ∈ C(Ω), the following problem (-△)^αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.

Fine Regularity of Solutions to the Dirichlet Problem Associated with the Regional Fractional Laplacian

In this paper, we study the H\"{o}lder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian $(-\Delta)^{\alpha}_{\Omega}$ on a bounded open set $\Omega\subset \mathds{R}$$(N\geq2)$ with $C^{1,1}$ boundary $\partial\Omega$. We prove that when $f\in L^{p}(\Omega)$, and $g\in C(\partial\Omega)$, the following problem $(-\Delta)^{\alpha}_{\Omega}u=f$ in $\Omega$, $u=g$ on $\partial\Omega$, admits a unique weak solution $u\in W^{\alpha,2}(\Omega)\cap C(\overline{\Omega})$, where $p>\frac{N}{2-2\alpha}$ and $\frac{1}{2}<\alpha<1$. To solve this problem, we consider it into two special cases, i.e., $g\equiv0$ on $\partial\Omega$ and $f\equiv0$ in $\Omega$. Finally, taking into account the preceding two cases, the general conclusion is drawn.