*(English)*Zbl 0932.35158

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

where ${B}_{j}(x,D)$ 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 $A(x,D)$ are not always formally selfadjoint. Let $\mathcal{L}$ be a sector in the complex plane with vertex at the origin. We assume that the operator $A$ is elliptic with parameter in $\mathcal{L}$. Let us consider the closed densely defined operator ${A}_{B}$ acting in ${L}^{2}\left(G\right)$, with domain

We see that in this case the resolvent set is nonvoid (contains all $\lambda \in \mathcal{L}$ with large $\left|\lambda \right|$) and the resolvent ${({A}_{B}-\lambda )}^{-1}$ 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 ${({A}_{B}-\lambda )}^{-1}$ in parameter $\lambda \in \mathcal{L}$. 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 $(\mathcal{L}\ni \lambda \to \infty )$ is given when $2mq>n$.