# Spectral asymptotics for Robin problems with a discontinuous coefficient

### Gerd Grubb

Copenhagen University, Denmark

## Abstract

The spectral behavior of the difference between the resolvents of two realizations of a second-order strongly elliptic symmetric differential operator $A$ defined by different Robin conditions $\chi u=b_1\gamma_0u$ and $\chi u=b_2\gamma_0u$, 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 $b_i$, showing that if $b_1$ and $b_2$ are in $L_\infty$, the s-numbers $s_j$ 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 $p>(n-1)/3$, under a hypothesis of boundedness of $b_i^{-1}$. Moreover, we show that if $b_1$ and $b_2$ are in $C^\varepsilon$ for some $\varepsilon >0$, with jumps at a smooth hypersurface, then $s_j j^{3/(n-1)}\to c$ for $j\to \infty$, with a constant defined from the principal symbol of $A$ and $b_2-b_1$. We also show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators of negative order interspersed with piecewise continuous functions.

## Cite this article

Gerd Grubb, Spectral asymptotics for Robin problems with a discontinuous coefficient. J. Spectr. Theory 1 (2011), no. 2, pp. 155–177

DOI 10.4171/JST/7