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 $\gamma _0u=0$ 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 $A_{\chi ,\Sigma _+}^{-1}-A_\gamma ^{-1}$.