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.).
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 |