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On the existence of nodal domains for elliptic differential operators. (English) Zbl 0537.35027
A bounded domain G is a nodal domain of an elliptic partial differential equation \((1)\quad Au=0\) of order 2m if there exists a non-trivial solution of (1), \(u\in \overset \circ W^ m_ 2(G)\cap W^{2m}_{2,loc}(G).\) The main result is that a self adjoint elliptic differential equation has rectangular nodal domains if the quadratic form of the equation takes on negative values. If the smallest point of the spectrum of a second order self-adjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the domain is shrinking.
Reviewer: S.Lenhart

35J30 Higher-order elliptic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI
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