论文标题
光谱分数拉普拉斯（Laplacian
Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions
论文作者
论文摘要
在本文中，我们讨论了方程$（-Δ）^s u = f $的正确设置的主题，其中$ 0 <s <1 $。整个空间中分数拉普拉斯的定义$ \ mathbb r^n $，$ n = 1,2,3 $是通过傅立叶变换理解的，例如，参见，例如，lischke et.al。 （J.Comp。Phys。，2020）。然而，真正的挑战代表了在有界域$ω$中提出此方程的情况，并且需要适当的边界条件以使相应问题的正确性。让我们在这里提到，直到最后几年，不均匀的边界数据已经忽略了。原因是在非本地设置中施加非零边界条件是高度不平凡的。至少有两个不同的定义对分数laplacian的定义，并且仍在进行有关它们的关系的持续研究。他们不等。我们研究的重点是光谱分数laplacian的新特征。当右手$ f $是dirac $δ$函数时，主要贡献之一是涉及的情况。为了比较光谱和Riesz公式中的溶液之间的差异，我们考虑了一个不均匀的分数Dirichlet问题。提供的理论分析由模型数值测试支持。
In this paper we discuss the topic of correct setting for the equation $(-Δ)^s u=f$, with $0<s <1$. The definition of the fractional Laplacian on the whole space $\mathbb R^n$, $n=1,2,3$ is understood through the Fourier transform, see, e.g., Lischke et.al. (J. Comp. Phys., 2020). The real challenge however represents the case when this equation is posed in a bounded domain $Ω$ and proper boundary conditions are needed for the correctness of the corresponding problem. Let us mention here that the case of inhomogeneous boundary data has been neglected up to the last years. The reason is that imposing nonzero boundary conditions in the nonlocal setting is highly nontrivial. There exist at least two different definitions of fractional Laplacian, and there is still ongoing research about the relations of them. They are not equivalent. The focus of our study is a new characterization of the spectral fractional Laplacian. One of the major contributions concerns the case when the right hand side $f$ is a Dirac $δ$ function. For comparing the differences between the solutions in the spectral and Riesz formulations, we consider an inhomogeneous fractional Dirichlet problem. The provided theoretical analysis is supported by model numerical tests.
