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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 ${ℝ}^{n}$ with $\left(n-1\right)$-dimensional boundary ${\Gamma }$, and $A\left(x,D\right)$ be a elliptic operator of order $2m$ on $G$. Consider the boundary value problem

$\begin{array}{cc}\hfill A\left(x,D\right)u\left(x\right)-\lambda u\left(x\right)& =f\left(x\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}G,\hfill \\ \hfill {B}_{j}\left(x,D\right)u\left(x\right)& ={g}_{j}\left(x\right)\phantom{\rule{1.em}{0ex}}\left(j=1,\cdots ,m\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma },\hfill \end{array}$

where ${B}_{j}\left(x,D\right)$ are partial differential operators of order ${m}_{j}<2m$ with coefficients defined only on ${\Gamma }$.

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

$D\left({A}_{B}\right)=\left\{u\in {W}_{2}^{2m}\left(G\right):{B}_{j}u=0\phantom{\rule{4pt}{0ex}}\left(j=1,\cdots ,m\right)\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma }\right\}·$

We see that in this case the resolvent set is nonvoid (contains all $\lambda \in ℒ$ with large $|\lambda |$) and the resolvent ${\left({A}_{B}-\lambda \right)}^{-1}$ is compact. Thus the spectrum of ${A}_{B}$ as an operator in ${L}^{2}\left(G\right)$ is discrete. If we consider the operators ${A}_{B}$, $p$ in ${L}^{p}\left(G\right)$ 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 $|{\lambda }_{j}|$ of the eigenvalues ${\lambda }_{j}$ by using the asymptotic behavior of the trace of the resolvent ${\left({A}_{B}-\lambda \right)}^{-1}$ in parameter $\lambda \in ℒ$. The main results are essentially Theorem 4.1 and Theorem 5.1. In these theorems the asymptotic behavior of the trace $\text{tr}\phantom{\rule{4.pt}{0ex}}R{\left(\lambda \right)}^{q}$ in $\lambda$ as $\left(ℒ\ni \lambda \to \infty \right)$ is given when $2mq>n$.

##### MSC:
 35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PD operators 35J40 Higher order elliptic equations, boundary value problems 47F05 Partial differential operators 58J05 Elliptic equations on manifolds, general theory
##### Keywords:
asymptotics; resolvent; eigenvalues