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Strict monotonicity of eigenvalues and unique continuation. (English) Zbl 0777.35042

This paper deals with the eigenvalue problem

Lu=μmuinΩ,u=0onΩ,

where Lu=- i,j=1 N D j (a ij D i u)+a 0 u is a uniformly elliptic operator in a bounded domain Ω N , with a ij =a ji L (Ω), 0a 0 . Moreover, the authors assume m¬0, a 0 , mL r (Ω) 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 ¬ means inequality a.e. together with strict inequality on a set of positive measure. Denote by μ 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 ¬m 2 , and let j 0 . If the eigenfunctions associated to μ j (m 1 ) enjoy the U.C.P., then μ j (m 1 )>μ j (m 2 ).

Theorem 2. Let m be a weight and let j + . If the eigenfunctions associated to μ j (m) do not enjoy the U.C.P., then there exists a weight m ^ with m¬m ^, such that, for some i 0 with μ i (m)=μ j (m), one has μ i (m)=μ i (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:
35P15Estimation of eigenvalues and upper and lower bounds for PD operators
35J15Second order elliptic equations, general
35B60Continuation of solutions of PDE