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Comparison theorems for eigenvalues

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Summary

Comparison theorems are proved relating the smallest positive eigenvalues λ and λ* of two operator equations of type Au=λBu and A*v=λ*B*v, respectively. Sufficient conditions on the operators are given which guarantee that λ⩽λ*, with special reference to the case that A and A* are elliptic differential operators. One novelty of the theory is that B is not required to be positive. Two general techniques are described: 1) A generalization of the classical minimum principle for eigenvalues, which is appropriate for selfadjoint elliptic operators A of arbitrary even order; and 2) A differential identity related to Picone's identity, appropriate for nonselfadjoint second order elliptic operators and strongly elliptic quasilinear systems.

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Entrata in Redazione il 24 maggio 1972.

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Allegretto, W., Swanson, C.A. Comparison theorems for eigenvalues. Annali di Matematica 99, 81–107 (1974) doi:10.1007/BF02413720

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Keywords

  • Differential Operator
  • Special Reference
  • Operator Equation
  • Elliptic Operator
  • General Technique