JournalsjstVol. 1, No. 2pp. 155–177

Spectral asymptotics for Robin problems with a discontinuous coefficient

  • Gerd Grubb

    Copenhagen University, Denmark
Spectral asymptotics for Robin problems with a discontinuous coefficient cover
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Abstract

The spectral behavior of the difference between the resolvents of two realizations of a second-order strongly elliptic symmetric differential operator AA defined by different Robin conditions χu=b1γ0u\chi u=b_1\gamma_0u and χu=b2γ0u\chi u=b_2\gamma_0u, can in the case where all coefficients are CC^\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 bib_i, showing that if b1b_1 and b2b_2 are in LL_\infty, the s-numbers sjs_j satisfy sjj3/(n1)Cs_j j^{3/(n-1)}\le C for all jj. This improves a recent result for A=ΔA=-\Delta by Behrndt et al., that jsjp<\sum_js_j ^p<\infty for p>(n1)/3p>(n-1)/3, under a hypothesis of boundedness of bi1b_i^{-1}. Moreover, we show that if b1b_1 and b2b_2 are in CεC^\varepsilon for some ε>0\varepsilon >0, with jumps at a smooth hypersurface, then sjj3/(n1)cs_j j^{3/(n-1)}\to c for jj\to \infty, with a constant defined from the principal symbol of AA and b2b1b_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