Vol. 30, No. 3, 1969

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A monotonicity principle for eigenvalues

Velmer B. Headley

Vol. 30 (1969), No. 3, 663–668

The smallest eigenvalue of certain boundary problems for second order linear elliptic partial differential equations increases to infinity as the domain in question shrinks to the empty set. The object of this note is to formulate and prove an analogous result for linear elliptic differential operators L of general even order. Specifically, let G(t) be a bounded domain in n-dimensional Euclidean space, and suppose that G(t) has thickness t (in a sense which will be precisely defined below). Let λ0(t) be the smallest eigenvalue of a boundary problem associated with L and G(t). It will be shown that λ0(t) increases to infinity as t tends to zero from the right.

Mathematical Subject Classification
Primary: 35.80
Received: 5 November 1968
Published: 1 September 1969
Velmer B. Headley