Spectral results for mixed problems and fractional elliptic operators

Abstract

In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_{+}$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{\chi ,{\Sigma }_{+}}$ in $L_2(\Omega )$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $\Omega \subset R^n$; here the boundary $\partial \Omega =\Sigma$ is partioned smoothly into $\Sigma ={\Sigma }_{-}\cup {\Sigma }_{+}$, the Dirichlet condition ${\gamma }_{0}u=0$ is imposed on ${\Sigma }_{-}$, and a Neumann or Robin condition $\chi u=0$ is imposed on ${\Sigma }_{+}$. It is shown that the Dirichlet-to-Neumann operator $P_{\gamma ,\chi }$ is principally of type $\frac{1}{2}$ with factorization index $\frac{1}{2}$, relative to ${\Sigma }_{+}$. The above theory allows a detailed description of $D(A_{\chi ,{\Sigma }_{+}})$ with singular elements outside of $H^{\frac{3}{2}}(\Omega )$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{\chi ,{\Sigma }_{+}}^{-1}-A_{\gamma }^{-1}$.

arXiv comment21 pages, introduction expanded with more references, small improvements in formulations

References in corpus (5)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators
• Spectral asymptotics for Robin problems with a discontinuous coefficient
• A short proof of Weyl's law for fractional differential operators
• Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Cited by in corpus (9)

• Numerical methods for nonlocal and fractional models
• Numerical Methods for Fractional Diffusion
• Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators
• On the maximum principle for higher-order fractional Laplacians
• Optimal operator preconditioning for pseudodifferential boundary problems
• Resolvents for fractional-order operators with nonhomogeneous local boundary conditions
• Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded Euclidean domain
• Semiclassical analysis of a nonlocal boundary value problem related to magnitude
• Neumann conditions for the higher order $s$-fractional Laplacian $(-\Delta {)}^{s}u$ with $s>1$