Linear symmetric hyperbolic systems with characteristic boundary

The author studies the mixed problem for the linear symmetric hyperbolic systems with maximally non-negative and characteristic boundary condition. Existence of a unique solution is proved inside a suitable class of functions of weighted Sobolev type which takes account of the loss of regularity in the normal direction to the characteristic boundary.     

The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ? R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω ₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.     

Noncommutative integrable systems on <Emphasis Type="Italic">b</Emphasis>-symplectic manifolds

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure.     

On singular partial differential boundary problems

Summary In the first part, the existence of a differential operator is established that is elliptic inside a domain and singular on the boundary. In the second part the invariance of the essential spectrum is established under a large class of perturbations of the original boundary problem. To Giovanni Sansone on his 70^th birthday. This research has partially been supported by a grant of the United States National Science Foundation.     

Symbolic dynamics of magnetic bumps

For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.     

Starshaped Locally Convex Hypersurfaces with Prescribed Curvature and Boundary

In this paper we find strictly locally convex hypersurfaces in \(\mathbb {R}^{n+1}\) with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show that any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.     

Problem of determining the permittivity in the stationary system of Maxwell equations

The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity ε from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant ε₀ only inside a compact domain Ω ? R ³ with a smooth boundary S. To find ε inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω₀ and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.     

On Markov diffusion processes with delayed reflection from boundaries of a segment

A continuous semi-Markov process with values in a closed interval is considered. This process coincides with a Markov diffusion process inside the interval. Thus, violation of the Markov property is only possible at the boundary of the interval. We prove a sufficient condition under which a semi-Markov process is Markov. We show that, in addition to Markov processes with instantaneous reflection from the boundary of the interval. there exists a class of Markov processes with delayed reflection from the boundary. Such a process has a positive average measure of time at which its trajectory belongs to the boundaries. This gives a different proof of a similar result by Gikhman and Skorokhod of 1968. Bibliography: 5 titles.     

Topological sensitivity analysis for some nonlinear PDE systems

The aim of the topological sensitivity analysis is to determine an asymptotic expansion of a design functional when creating a small hole inside the domain. In this work, such an expansion is obtained for a certain class of nonlinear PDE systems of order 2 in dimensions 2 and 3 with a Dirichlet condition prescribed on the boundary of an arbitrarily shaped hole. Some examples of such operators are presented.     

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.     

Diffusion processes with delay at the endpoints of a segment

A continuous semi-Markov process with a segment as the range of values is considered. This process coincides with a diffusion process inside the segment, i.e., up to the first hitting time of the boundary of the segment and at any time when the process leaves the boundary. The class of such processes consists of Markov processes with reflection at the boundaries (instantaneously or with a delay) and semi-Markov processes with intervals of constancy on some boundary. We derive conditions of existence of such a process in terms of a semi-Markov transition generating function on the boundary. The method of imbedded alternating renewal processes is applied to find a stationary distribution of the process. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 284–297.     

Weighted Lipschitz spaces and their analytic characterizations

Weighted Lipschitz spaces are studied where the growth function varies, i.e., $$|f(x + h) - f(x)| \leqslant C\rho (h)$$ and where the growth function may also depend on x. Connections between a functionf defined on the complex unit circle and its analytic extensionF to the inside are given. In particular, theory relating the Lipschitz growth of the boundary function and derivative growth of the analytic function is developed. An atomic decomposition of the boundary values of a weighted Besov space is also obtained.     

一类Hamiltion系统的Abel积分的零点个数估计

In this paper, we discuss the estimation of the number of zeros of the Abelian integral for the quadratic system which has a periodic region with a parabola and a straight line as its boundary when we perturb the system inside the class of all polynomial systems of degree n. The main result is that the upper bound for the number of zeros of the Abelian integral associated to this system is 3n-1.     

Reconstruction of inclusions in an elastic body

We consider the reconstruction of elastic inclusions embedded inside of a planar region, bounded or unbounded, with isotropic inhomogeneous elastic parameters by measuring displacements and tractions at the boundary. We probe the medium with complex geometrical optics solutions having polynomial-type phase functions. Using these solutions we develop an algorithm to reconstruct the exact shape of a large class of inclusions including star-shaped domains and we implement numerically this algorithm for some examples.     

Stress obtained by interpolation methods for a boundary value problem in linear viscoelasticity

We will solve the following boundary value problem in linear viscoelasticity: given the value of the stress on (a part of) the boundary of the domain find the stress in the whole body at all positive times. Using a state space setting we show that the stress field inside the body can be obtained from the boundary stress by a variation-of-parameters formula involving an analytic semigroup. The relation between the regularities of the boundary stress and the stress inside the body is therefore characterized by the well-known and rich regularity theory for analytic semigroups.     

Energy estimates for solutions of the mixed problem for linear second-order hyperbolic equations

A mixed problem for a linear second-order hyperbolic equation with antidissipation inside the domain and dissipation on a part of the boundary is considered. It is proved that for certain relations between the antidissipation inside the domain and the dissipation on the part of the boundary, the energy of the system exponentially decreases, whereas for sufficiently large antidissipation inside the domain the boundary dissipation has no effect on the energy of the system; in this case the energy remains unbounded. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 483–488, April, 1996.     

On Asymptotic Effects of Boundary Perturbations in Exponentially Shaped Josephson Junctions

A parabolic integro differential operator \(\mathcal{L}\) , suitable to describe many phenomena in various physical fields, is considered. By means of equivalence between \(\mathcal{L}\) and the third order equation describing the evolution inside an exponentially shaped Josephson junction (ESJJ), an asymptotic analysis for (ESJJ) is achieved, explicitly evaluating, boundary contributions related to the Dirichlet problem.     

Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains

We establish the existence of the classical solution for the pressure-gradient equation in a non-smooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.     

Harnack-type mass bounds and bernstein theorems for area-minimizing flat chains modulo v.

For an area-minimizing flat chain modulo v with no boundary inside the unit ball, an absolute upper bound is given for the amount of area inside a shrunken ball. Such Harnack-type estimates lead to generalizations of area inside a shrunken ball. Such Harnack-type estimates lead to generalizations of Bernstein's Theorem. For example, for n?5, a 2-dimensional, area-minimizing locally flat chain modulo 2 without boundary in IRⁿ which has at least 1 singularity must consist of 2 orthogonal planes.