Spectral Fractional Laplacian with Inhomogeneous Dirichlet Data: Questions, Problems, Solutions

In this paper we discuss the topic of correct setting for the equation $(-\Delta )^s u=f$, 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 $\Omega$ 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 $\delta$ 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.