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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 $$Lu=-\sum^ N_{i,j=1} D_ j(a_{ij} D_ i u)+a_ 0 u$$ is a uniformly elliptic operator in a bounded domain $$\Omega\subset\mathbb{R}^ N$$, with $$a_{ij}=a_{ji}\in L^ \infty(\Omega)$$, $$0\leq a_ 0$$. Moreover, the authors assume $$m\not\equiv 0$$, $$a_ 0$$, $$m\in L^ r(\Omega)$$ 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 $$\leq\not\equiv$$ means inequality a.e. together with strict inequality on a set of positive measure. Denote by $$\mu_ j(m)$$ the eigenvalues of $$L$$ with respect to the weight $$m$$. The main results of the paper are:
Theorem 1. Let $$m_ 1$$ and $$m_ 2$$ be two weights with $$m_ 1\leq\not\equiv m_ 2$$, and let $$j\in\mathbb{Z}_ 0$$. If the eigenfunctions associated to $$\mu_ j(m_ 1)$$ enjoy the U.C.P., then $$\mu_ j(m_ 1)>\mu_ j(m_ 2)$$.
Theorem 2. Let $$m$$ be a weight and let $$j\in \mathbb{Z}_ +$$. If the eigenfunctions associated to $$\mu_ j(m)$$ do not enjoy the U.C.P., then there exists a weight $$\widehat m$$ with $$m\leq\not\equiv \widehat m$$, such that, for some $$i\in\mathbb{Z}_ 0$$ with $$\mu_ i(m)=\mu_ j(m)$$, one has $$\mu_ i(m)=\mu_ i(\widehat m)$$.
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

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35J15 Second-order elliptic equations 35B60 Continuation and prolongation of solutions to PDEs
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