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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 $\bbfR^n$ with $(n-1)$-dimensional boundary $\Gamma$, and $A(x,D)$ be a elliptic operator of order $2m$ on $G$. Consider the boundary value problem $$\align A(x,D)u(x)-\lambda u(x) & = f(x)\quad\text{in }G,\\ B_j(x, D)u(x) & = g_j(x)\quad (j= 1,\dots, m)\quad\text{ on }\Gamma,\endalign$$ 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 ${\cal L}$ be a sector in the complex plane with vertex at the origin. We assume that the operator $A$ is elliptic with parameter in ${\cal L}$. Let us consider the closed densely defined operator $A_B$ acting in $L^2(G)$, with domain $$D(A_B)= \{u\in W^{2m}_2(G): B_ju= 0\ (j=1,\dots, m)\text{ on }\Gamma\}.$$ We see that in this case the resolvent set is nonvoid (contains all $\lambda\in{\cal L}$ with large $|\lambda|$) and the resolvent $(A_B-\lambda)^{-1}$ 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 $|\lambda_j|$ of the eigenvalues $\lambda_j$ by using the asymptotic behavior of the trace of the resolvent $(A_B- \lambda)^{-1}$ in parameter $\lambda\in{\cal L}$. The main results are essentially Theorem 4.1 and Theorem 5.1. In these theorems the asymptotic behavior of the trace $\text{tr }R(\lambda)^q$ in $\lambda$ as $({\cal L}\ni\lambda\to \infty)$ is given when $2mq> n$. For the entire collection see [Zbl 0882.00015].

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