Spectral asymptotics for Robin problems with a discontinuous coefficient

TL;DR

This paper investigates the spectral asymptotics of differences in resolvents for elliptic operators with Robin boundary conditions, extending previous smooth coefficient results to nonsmooth cases and providing sharper estimates.

Contribution

It extends spectral asymptotic analysis to nonsmooth Robin coefficients and derives sharper eigenvalue decay estimates using Krein resolvent formulas.

Findings

s-numbers decay rate improved for nonsmooth coefficients

Sharper asymptotic behavior when coefficients are Hölder continuous

Principal spectral asymptotics extended to piecewise continuous functions

Abstract

The spectral behavior of the difference between the resolvents of two realizations $\tilde A_1$ and $\tilde A_2$ of a second-order strongly elliptic symmetric differential operator $A$, defined by different Robin conditions $\nu u=b_1\gamma_0u$ and $\nu u=b_2\gamma_0u$, can in the case where all coefficients are $C^\infty$ be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth $b_i$. Using a Krein resolvent formula, we show that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ of $(\tilde A_1 -\lambda)^{-1}-(\tilde A_2 -\lambda)^{-1}$ satisfy $s_j j^{3/(n-1)}\le C$ for all $j$; this improves a recent result for $A=-\Delta $ by Behrndt et al., that $\sum_js_j ^p<\infty$ for $p>(n-1)/3$. A sharper estimate is obtained when $b_1$ and $b_2$ are in $C^\epsilon$ for some $\epsilon >0$, with jumps at a smooth hypersurface, namely that $s_j j^{3/(n-1)}\to c$ for $j\to \infty$, with a constant $c$ defined from the principal symbol of $A$ and $b_2-b_1$. As an auxiliary result we show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators interspersed with piecewise continuous functions.