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.).