Su alcune questioni di Calcolo delle Variazioni: omogeneizzazione in domini perforati con condizioni di Neumann e rilassamento
Complete asymptotic expansions for eigenvalues of Dirichlet laplacian in thin three-dimensional rods
We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions...
Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods
We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions...
Bound states of a converging quantum waveguide
We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 − where > 0 is a small real parameter, the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + (√ε). We will present a sufficient condition for the existence of a weakly coupled bound state below , the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when...
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