On the principal eigenvalue of a periodic-parabolic operator. (English) Zbl 0563.35033
Let \(\Omega \subset {\mathbb{R}}^ N\) be a bounded domain, and let \(L:=\partial /\partial t+A(x,t,D)\) be a uniformly parabolic differential expression which is T-periodic in t. The authors study the eigenvalue problem (*) \(Lu=\lambda mu\) in a space of Hölder continuous T-periodic functions on \({\bar \Omega}\times {\mathbb{R}}\), where \(m\neq 0\) is a (not necessarily positive) weight function.
Main result: (*) has a positive eigenvalue \(\lambda_ 1(m)\) having a positive eigenfunction u if and only if \(\int^{T}_{0}(\max_{x}m(x,t))dt\) is positive. The result generalizes previous results for (a) \(m=1\), (b) elliptic eigenvalue problems with indefinite weight functions.
Main result: (*) has a positive eigenvalue \(\lambda_ 1(m)\) having a positive eigenfunction u if and only if \(\int^{T}_{0}(\max_{x}m(x,t))dt\) is positive. The result generalizes previous results for (a) \(m=1\), (b) elliptic eigenvalue problems with indefinite weight functions.
Reviewer: J.Voigt
MSC:
35K10 | Second-order parabolic equations |
47F05 | General theory of partial differential operators |
35P05 | General topics in linear spectral theory for PDEs |
Keywords:
principal eigenvalue; parabolic differential expression; periodic; positive eigenvalue; positive eigenfunction; indefinite weight functionsReferences:
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