Lavrent'ev A. S.
Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1467–1475
We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as $ε → 0$. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as $ε → 0$ and associated with natural forms of vibrations localized in the domain of perturbation of density.
Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1206–1212
We solve the problem of construction of a two-parame to either a multiplicative or a coordinatewise two-parameter semigroup. The construction is carried out on the basis of the “initial family of kernels.”
Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 57-65
We consider multiparameter semigroups of two types (multiplicative and coordinatewise) and resolvent operators associated with such semigroups. We prove an alternative version of the Hille-Yosida theorem in terms of resolvent operators. For simplicity of presentation, we give statements and proofs for two-parameter semigroups.