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On the existence of nodal domains for elliptic differential operators

Published online by Cambridge University Press:  14 November 2011

E. Müller-Pfeiffer
E. Müller-Pfeiffer
Affiliation:
Erfurt, G.D.R
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Synopsis

We prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 3-4 , 1983 , pp. 287 - 299
Copyright
Copyright © Royal Society of Edinburgh 1983

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