On the spectral Stefan–Florin problem with classical boundary condition

Abstract

The purpose of this work is to study the spectral properties of the problem of transmission arising after the linearization of two-phase problems of Stefan and Florin with classical boundary condition on a small time interval. With the help of the operator methods of mathematical physics, a boundary-value problem is reduced to the study of the spectrum of a weakly perturbed compact self-adjoint operator in a Hilbert space. On the basis of the theorems of M. V. Keldysh and V. B. Lidskii, we have established the basis property of the system of eigen- and associated elements by Abel–Lidskii in some Hilbert space. It is proved that the spectrum is discrete with the single limiting point at infinity. It is situated on the positive semiaxis or, except for a finite number of eigenvalues, in the aperture angle ε. The growth of the moduli of eigenvalues is estimated, and some asymptotic formulas are obtained.