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Strict monotonicity of eigenvalues and unique continuation. (English) Zbl 0777.35042
This paper deals with the eigenvalue problem $$Lu=\mu mu\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega,$$ where $Lu=-\sum\sp N\sb{i,j=1} D\sb j(a\sb{ij} D\sb i u)+a\sb 0 u$ is a uniformly elliptic operator in a bounded domain $\Omega\subset\bbfR\sp N$, with $a\sb{ij}=a\sb{ji}\in L\sp \infty(\Omega)$, $0\le a\sb 0$. Moreover, the authors assume $m\not\equiv 0$, $a\sb 0$, $m\in L\sp r(\Omega)$ for some $r>N/2$. The authors investigate the monotone dependence on the weight $m$ of the eigenvalues of the problem. For stating their results, we need the notion of “unique continuation property”: A family of functions is said to enjoy the unique continuation property (U.C.P.), if no function, besides possibly the zero function, vanishes on a set of positive measure. The notation $\le\not\equiv$ means inequality a.e. together with strict inequality on a set of positive measure. Denote by $\mu\sb j(m)$ the eigenvalues of $L$ with respect to the weight $m$. The main results of the paper are: Theorem 1. Let $m\sb 1$ and $m\sb 2$ be two weights with $m\sb 1\le\not\equiv m\sb 2$, and let $j\in\bbfZ\sb 0$. If the eigenfunctions associated to $\mu\sb j(m\sb 1)$ enjoy the U.C.P., then $\mu\sb j(m\sb 1)>\mu\sb j(m\sb 2)$. Theorem 2. Let $m$ be a weight and let $j\in \bbfZ\sb +$. If the eigenfunctions associated to $\mu\sb j(m)$ do not enjoy the U.C.P., then there exists a weight $\widehat m$ with $m\le\not\equiv \widehat m$, such that, for some $i\in\bbfZ\sb 0$ with $\mu\sb i(m)=\mu\sb j(m)$, one has $\mu\sb i(m)=\mu\sb i(\widehat m)$. Furthermore, the last section of the paper discusses the relationship between U.C.P. and the strong unique continuation property (S.U.C.P.).
Reviewer: M.Lesch

35P15Estimation of eigenvalues and upper and lower bounds for PD operators
35J15Second order elliptic equations, general
35B60Continuation of solutions of PDE
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