This paper deals with the eigenvalue problem
where is a uniformly elliptic operator in a bounded domain , with , . Moreover, the authors assume , , for some .
The authors investigate the monotone dependence on the weight 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 means inequality a.e. together with strict inequality on a set of positive measure. Denote by the eigenvalues of with respect to the weight . The main results of the paper are:
Theorem 1. Let and be two weights with , and let . If the eigenfunctions associated to enjoy the U.C.P., then .
Theorem 2. Let be a weight and let . If the eigenfunctions associated to do not enjoy the U.C.P., then there exists a weight with , such that, for some with , one has .
Furthermore, the last section of the paper discusses the relationship between U.C.P. and the strong unique continuation property (S.U.C.P.).