Summary
This paper is concerned with the monotonicity of the first eigenvalue λ1 (D) of (1) as a functional of the domainD.
For 0<k<∞ we prove that λ1(D) is increasing withD in several cases, but a counterexample is given showing that this is not true in general. In the same cases we prove that, for −∞<k<0, λ1(D) decreases whenD increases.
Zusammenfassung
Diese Arbeit behandelt die Monotonie des ersten Eigenwertes λ1(D) von (1) als Funktional des GebietesD. Für 0<k<∞ beweisen wir, dass λ1(D) in mehreren Fällen mitD wächst; aber ein Gegenbeispiel zeigt, dass dies nicht allgemein gilt. In denselben Fällen beweisen wir, dass λ1(D) für −∞<k<0 bei wachsendemD abnimmt.
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References
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Technical Report TR-75-13.
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Bareket, M. On the domain monotonicity of the first eigenvalue of a boundary value problem. Journal of Applied Mathematics and Physics (ZAMP) 27, 487–491 (1976). https://doi.org/10.1007/BF01594905
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DOI: https://doi.org/10.1007/BF01594905