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Weakly smooth nonselfadjoint spectral elliptic boundary problems. (English) Zbl 0932.35158
Demuth, Michael (ed.) et al., Spectral theory, microlocal analysis, singular manifolds. Berlin: Akademie Verlag. Math. Top. 14, 138-199 (1997).

Let G be a bounded domain in n with (n-1)-dimensional boundary Γ, and A(x,D) be a elliptic operator of order 2m on G. Consider the boundary value problem

A(x,D)u(x)-λu(x)=f(x)inG,B j (x,D)u(x)=g j (x)(j=1,,m)onΓ,

where B j (x,D) are partial differential operators of order m j <2m with coefficients defined only on Γ.

The paper treats the case that the coefficients of the operators are not always smooth, and moreover the operators A(x,D) are not always formally selfadjoint. Let be a sector in the complex plane with vertex at the origin. We assume that the operator A is elliptic with parameter in . Let us consider the closed densely defined operator A B acting in L 2 (G), with domain

D(A B )={uW 2 2m (G):B j u=0(j=1,,m)onΓ}·

We see that in this case the resolvent set is nonvoid (contains all λ with large |λ|) and the resolvent (A B -λ) -1 is compact. Thus the spectrum of A B as an operator in L 2 (G) is discrete. If we consider the operators A B , p in L p (G) spaces, then they are spectrally equivalent (Section 3). The concern of the paper is to investigate the spectral property of the operator A B and to get a result on the asymptotic behavior in the absolute value |λ j | of the eigenvalues λ j by using the asymptotic behavior of the trace of the resolvent (A B -λ) -1 in parameter λ. The main results are essentially Theorem 4.1 and Theorem 5.1. In these theorems the asymptotic behavior of the trace trR(λ) q in λ as (λ) is given when 2mq>n.


MSC:
35P20Asymptotic distribution of eigenvalues and eigenfunctions for PD operators
35J40Higher order elliptic equations, boundary value problems
47F05Partial differential operators
58J05Elliptic equations on manifolds, general theory