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A variational problem with impurity set. (English) Zbl 1156.35069

The main part of this work is concerned with the somewhat generalised eigenvalue problem of Helmholtz equations defined on a bounded, connected and open set in N-dimensional Euclidean space. This set is assumed to be Lebesgue measurable. Furthermore, it is supposed that it is expressible as the union of two disjoint measurable sets one of which is called the impurity set. The eigenvalues on the complementary set and the impunity set differ by a given positive factor that is bounded above by the ratio of measures of the impurity set and its complement. The relevant eigenvalue is evaluated through a modified form of the classical variational principle in which the denominator is replaced by the difference of some integrals on the complementary and impurity sets.
The authors employ ingeniously the theory of elliptic differential equations and various norm inequalities involving Lebesgue and Sobolev spaces to obtain some estimates of the eigenvalue which depends significantly on the location and the measure of the impurity set. Moreover, the properties of the minimum of the eigenvalue with respect to the impurity set are investigated in details. The results obtained are utilised to study Ginzburg-Landau equations modelling a superconducting material occupying a region in N-dimensional Euclidean space with a normal impurity inclusion.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J20 Variational methods for second-order elliptic equations
35A15 Variational methods applied to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
82D55 Statistical mechanics of superconductors

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References:

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