SPECTRAL PROPERTIES AND DISTRIBUTION OF
EIGENVALUES OF NON–SELF-ADJOINT ELLIPTIC
DIFFERENTIAL OPERATORS
Reza Alizadeh1, Ali Sameripour2
1,2 Mathematics Dept., Lorestan University
Khoramabad, IRAN

Abstract

Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary $\partial\Omega$. In this article, we investigate the spectral properties of a non-selfadjoint elliptic differential operator $(Pu)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x) Q(x)u'_{x_{i}}(x)\right)'_{x_{j}}$, acting in the Hilbert space $H_{\ell}=L^{2}{(\Omega)}^{\ell}$ with Dirichlet-type boundary conditions. Here $c\vert s\vert^{2} \leq \sum^{n}_{i,j=1}a_{ij}(x)s_{i} \overline{s_{j}}\;\;\;(s=... ...j}(x)= \overline{a_{ji}(x)}\in C^{2}(\overline{\Omega}), \;\; 0 \leq \alpha < 1$. Furthermore, suppose that $\omega \in {{C}^{1}}\left( \bar{\Omega },\mathcal{R} \right)$ and this function is a positive function and called the weight function and $Q(x) \in C^{2}(\overline{\Omega},\;End\; {\bf C}^{\ell})$ such that for each $x\in \overline{\Omega}$ the matrix function $Q(x)$ has non-zero simple eigenvalues ${{\mu }_{j}}(x)\in {{C}^{2}}(\bar{\Omega },{\bf C})\ \ (1\le j\le \ell )$ lie in the $\psi_{\theta_1\theta_2}$ and real numbers and here $\psi_{\theta_1\theta_2}=\{z \in {\bf C}:\;\pi/2<\theta_1 \leq \vert arg\;z\vert \leq \theta_2<\pi\},$

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 1
Year: 2021

DOI: 10.12732/ijam.v34i1.11

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