Spectral results for mixed problems and fractional elliptic operators

TL;DR

This paper derives spectral asymptotics for fractional elliptic operators and applies the results to analyze mixed boundary problems, including the behavior of eigenfunctions and the spectral difference of related operators.

Contribution

It provides Weyl type spectral asymptotics for fractional powers of elliptic operators and characterizes the domain and spectral properties of mixed boundary value problems.

Findings

Weyl asymptotic formulas for fractional elliptic operators

Description of eigenfunction regularity and domain

Spectral asymptotics for the Krein resolvent difference

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 $\gamma _0u=0$ 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 $\frac12$ with factorization index $\frac12$, relative to $\Sigma _+$. The above theory allows a detailed description of $D(A_{\chi ,\Sigma _+})$ with singular elements outside of $H^{\frac32}(\Omega )$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{\chi ,\Sigma _+}^{-1}-A_\gamma ^{-1}$.