Asymptotic behavior of the spectrum of certain problems,connected with the oscillation of fluids

One obtains the principal term of the asymptotics of the spectrum in a series of problems of the theory of small oscillations of fluids, filling partially a container. First one discusses a certain problem of general character, the nonlocal Steklov type problem. The considered problems are connected with the oscillations of a capillary ideal (or capillary stratified) fluid, of a heavy viscous fluid, of a capillary viscous fluid, and with the oscillations of a fluid in container with an elastic bottom.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 158–164, 1986.The author is deeply grateful to her scientific advisor M. Sh. Birman. The author is grateful also to N. D. Kopachevskii for his interest in the paper.     

Adaptive detection of a high-variable function

A major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality:” Rates of accuracy in estimation and separation rates in detection problems behave poorly as the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or H?lder balls.     

Minimax nonparametric testing in a problem related to the radon transform

We consider the detection problem of a two-dimensional function from noisy observations of its integrals over lines. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are non-adaptive. We also construct a minimax rate-optimal adaptive test of rather simple structure.     

Asymptotics of the spectra of variational problems on the solutions of a homogeneous elliptic equation,under the presence of a constraint on part of the boundary

One obtains the principal term of the asymptotics of the spectrum of the ratio of two differential quadratic forms, considered in a bounded domain on the solutions of a homogeneous, regularly elliptic equation, under the presence of additional conditions on part of the boundary. This condition has the form of a pseudodifferential constraint and it is imposed on the Dirichlet data of the solutions. On the leading symbol of the corresponding pseudodifferential operator on the boundary, one imposes a nonsingularity conditioa.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 84–97, 1984.In conclusion, the author expresses her gratitude to her scientific adviser M. Sh. Birman for his assistance and also to M. Z. Solomyak for his interest in the paper and for a series of useful consultations.     

In Memory of Sergei Yuryevich Slavyanov

Theoretical and Mathematical Physics -     

Asymptotics of the spectra of variational problems on the solutions of an elliptic equation in a domain with a piecewise-smooth boundary

In a bounded domain with a piecewise-smooth boundary one considers the ratio of differential quadratic forms on the solutions of a homogeneous, strongly elliptic equation. One obtains the principal term of the asymptotics of the spectrum of the compact operator defined by this ratio.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 179–183, 1985.The author is sincerely grateful to M. Sh. Birman for guidance.     

Homogenization of a multidimensional periodic elliptic operator in a neighborhood of the edge of an internal gap

The homogenization procedure for a multidimensional periodic Schrödinger operator near the edge of an internal gap is discussed. We obtain an approximation for the resolvent in the small period limit with respect to the operator norm in L₂(?^d). This approximation contains oscillations but in a simpler form than the resolvent of the initial operator. Bibliography: 8 titles.     

Homogenization of the parabolic cauchy problem in the Sobolev class H<Superscript>1</Superscript>(R<Superscript>d</Superscript>)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in R^d is studied. The coefficients are assumed to be periodic in R^d with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(R^d) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H ¹(R^d) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H ¹(R^d) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.