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Evgeny Lakshtanov

Publications and source records attributed to Evgeny Lakshtanov.

33 records · Page 2Linked to original sources
arXiv2013

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction $\sqrt{n(x)}$ is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that $n(x)-1$ does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued $n(x)$.

math-ph↗
arXiv2013

Weyl type bound on positive Interior Transmission Eigenvalues

This paper contains a lower bound of the Weyl type on the counting function of the positive eigenvalues of the interior transmission eigenvalue problem which justifies the existence of an infinite set of positive interior transmission eigenvalues. We consider the classical transmission problem as well as the case where the inhomogeneous medium contains an obstacle. One of the essential components of the proof is an estimate for the D-t-N operator for the Helmholtz equation for positive $λ$ that replaces the standard parameter-elliptic estimate valid outside of the positive semi-axis.

math.SP↗
arXiv2012

Bounds on positive interior transmission eigenvalues

The paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.

math-ph↗
arXiv2012

Return Radius and volume of recrystallized material in Ostwald Ripening

Within the framework of the LSW theory of Ostwald ripening the amount of volume of the second (solid) phase that is newly formed by recrystallization is investigated. It is shown, that in the late stage, the portion of the newly generated volume formed within an interval from time $t_0$ to $t$ is a certain function of $t/t_0$ and an explicit expression of this volume is given. To achieve this, we introduce the notion of the {\it return radius} $r(t,t_0)$, which is the unique radius of a particle at time $t_0$ such that this particle has -- after growing and shrinking -- the same radius at time $t$. We derive a formula for the return radius which later on is used to obtain the newly formed volume. Moreover, formulas for the growth rate of the return radius and the recrystallized material at time $t_0$ are derived.

math-ph↗
arXiv2012

Isotope uptake dynamics in the Ostwald ripening model of recrystallization

The article is withdrawn since we decided not to publish it in the present form but to divide it in two parts. One of these parts concerns the mathematical aspects of Return Radius calculation and the rewritten article is already available arXiv:1201.4492 The second part will discuss the modelling of the Isotope Uptake and is still in preparation

math-ph↗
arXiv2012

A priori estimates for high frequency scattering by obstacles of arbitrary shape

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number $k\to \infty$. This bound of the total cross section is sharp.

math-ph↗
arXiv2011

Finiteness in the Card Game of War

The game of war is one of the most popular international children's card games. In the beginning of the game, the pack is split into two parts, then on each move the players reveal their top cards. The player having the highest card collects both and returns them to the bottom of his hand. The player left with no cards loses. Those who played this game in their childhood did not always have enough patience to wait until the end of the game. A player who has collected almost all the cards can lose all but a few cards in the next 3 minutes. That way the children essentially conduct mathematical experiments observing chaotic dynamics. However, it is not quite so, as the rules of the game do not prescribe the order in which the winning player will put his take to the bottom of his hand: own card, then rival's or vice versa: rival's card, then own. We provide an example of a cycling game with fixed rules. Assume now that each player can seldom but regularly change the returning order. We have managed to prove that in this case the mathematical expectation of the length of the game is finite. In principle it is equivalent to the graph of the game, which has got edges corresponding to all acceptable transitions, having got the following property: from each initial configuration there is at least one path to the end of the game.

math.DS↗
arXiv2011

Examples of Admissible Simplification of Mathematical Theories

"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some examples from our experience, when we were able to simplify some serious mathematical models to make them understandable by children, preserving both aesthetic and intellectual value. The latter is in particularly measured by whether a given simplification allows setting a sufficient list of problems feasible for school students.

math.HO↗
arXiv2011

Ellipticity in the interior transmission problem in anisotropic media

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.

math-ph↗
arXiv2010

Resonance regimes of scattering by small bodies with impedance boundary conditions

The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the solutions of the scattering problem. The study includes the case when $k=0$ is an eigenvalue or a resonance. The transformation from the impedance to the Dirichlet boundary condition as impedance grows is described. A relation between poles and zeroes of the scattering matrix in the non-self adjoint case is established. The results are applied to a problem of scattering by an obstacle with a springy coating. The paper describes the dependence of the impedance on the properties of the material, that is on forces due to the deviation of the boundary of the obstacle from the equilibrium position.

math-ph↗
arXiv2009

"Minesweeper" and spectrum of discrete Laplacians

The paper is devoted to a problem inspired by the "Minesweeper" computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of "Minesweeper" but does not require a computer.

cs.DM↗
arXiv2009

Spectral properties of the Dirichlet-to-Neumann operator for exterior Helmholtz problem and its applications to scattering theory

We prove that the Dirichlet-to-Neumann operator (DtN) has no spectrum in the lower half of the complex plane. We find several application of this fact in scattering by obstacles with impedance boundary conditions. In particular, we find an upper bound for the gradient of the scattering amplitude and for the total cross section. We justify numerical approximations by providing bounds on difference between theoretical and approximated solutions without using any a priory unknown constants.

math-ph↗
arXiv2008

Sobre o papel dos Departamentos de Matemática na vida e desenvolvimento da comunidade

The objective of this article is to stimulate discussions in mathematical society about the role of mathematical departments in the life of the community. University community is the center of knowledge and promotes the intellectual development. However, this is questioned today because of its reduced participation on the global learning process as there are many other components.

math.HO↗