We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-\Delta)^s u =f$ in $\Omega$, $u \equiv 0$ on $\mathbb{R}^N\setminus \Omega$ in an open bounded set $\Omega \subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $\Omega$ is of class $C^2$ and $f \in C^{\alpha}(\bar{\Omega})$ for some $\alpha>0$, we show that the map $(0,1) \to L^\infty(\Omega)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions on $f$ and $\Omega$. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case $s=1$, i.e., for the local Dirichlet problem $-\Delta u = f$ in $\Omega$, $u \equiv 0$ on $\partial \Omega$.

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通过对数拉普拉斯算子对分数泊松问题的新看法

我们分析了对分数泊松问题族 $(-\Delta)^su =f$ in $\Omega$, $u \equiv 0$ on $\mathbb{R} 的解 $u_s$ 的 $s$-依赖性^N\setminus \Omega$ 在开有界集合 $\Omega \subset \mathbb{R}^N$, $s \in (0,1)$。在 $\Omega$ 属于 $C^2$ 类并且 $f \in C^{\alpha}(\bar{\Omega})$ 对于某些 $\alpha>0$ 的情况下，我们表明地图$(0,1) \to L^\infty(\Omega)$, $s\mapsto u_s$ 属于 $C^1$ 类，我们根据对数拉普拉斯算子来表征导数 $\partial_s u_s$ $f$。作为推论，我们在 $f$ 和 $\Omega$ 的适当假设下推导出解映射 $s \mapsto u_s$ 的逐点单调性。此外，我们在任意有界域上为相应的 Green 算子推导出显式边界，即使在 $s=1$ 的情况下也是新的，即，