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Alexander I. Nazarov

Publications and source records attributed to Alexander I. Nazarov.

46 records · Page 3Linked to original sources
arXiv2013

Oblique derivative problem for non-divergence parabolic equations with discontinuous in time coefficients

We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in $t$ coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class ${\cal C}^{1,δ}$, $δ\in (0,1]$, then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary.

math.AP↗
arXiv2012

On formula of regularized traces II

We obtain a simple formula for the first-order trace of a regular differential operator on a segment perturbated by a multiplication operator. The main analytic ingredient of the proof is an improvement of the Tamarkin equiconvergence theorem.

math.SP↗
arXiv2011

A centennial of the Zaremba--Hopf--Oleinik Lemma

We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure the Hopf--Oleinik Lemma for solutions to hold true. We also touch the gradient estimates for solutions at the boundary.

math.AP↗
arXiv2010

Small Ball Probabilities for Smooth Gaussian fields and Tensor Products of Compact Operators

We find the logarithmic $L_2$-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein--Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper \cite{KNN}, where the regular behavior at infinity of marginal eigenvalues was assumed.

math.PR↗