A two-dimensional inverse problem for an integro-differential equation of electrodynamics

An integro-differential equation corresponding to a two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the x ₃ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integro-differential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain Ω ? ?². To find this coefficient inside Ω, we use information on the solution of the corresponding direct problem on the boundary of Ω on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.     

Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation

For an integrodifferential equation corresponding to a two-dimensional viscoelastic problem, we study the problem of defining the spatial part of the kernel involved in the integral term of the equation. The support of the sought function is assumed to belong to a compact domain Ω. As information for solving this inverse problem, the traces of the solution to the direct Cauchy problem and its normal derivative are given for some finite time interval on the boundary of Ω. An important feature in the statement of the problem is the fact that the solution of the direct problem corresponds to the zero initial data and a force impulse in time localized on a fixed straight line disjoint with Ω. The main result of the article consists in obtaining a Lipschitz estimate for the conditional stability of the solution to the inverse problem under consideration.     

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.     

On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions . We assume that may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ². We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles. Dedicated to O. A. Ladyzhenskaya on the occasion of her 70^th birthday. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 159–178, 1992. Translated by E. V. Frolova.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W ₂ ² (Ω) of the equation Δ _x ² u = f with the boundary conditions u = Δ_xu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ C^∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel and co-kernel of the operators and determining the solution u ∈ W ₂ ² (Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a nonnegative function f ∈ L₂(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property remain open. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

Oseen and Stokes asymptotics for the problem on stationary motion of two immiscible liquids

The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω₁ ⊂ ℝ³, whereas the other occupies the exterior domain Ω₂=ℝ⁴∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem describing a uniform drop in an infinite liquid. Bibliography: 10 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 208–238.     

The initial boundary-value problem with a free surface condition for the penalized equations of aqueous solutions of polymers

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (ℝ⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω⊂ℝ³. Bibliography: 25 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207. Translated by N. A. Karazeeva.     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L ²-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.     

A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ?B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ?B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.     

Sharp regularity estimates for solutions of the wave equation and their traces with prescribed neumann data

In this paper the regularity properties of second-order hyperbolic equations defined over a rectangular domain Θ with boundary Γ under the action of a Neumann boundary forcing term inL ² (0,T;H ^1/4 (Γ)) are investigated. With this given boundary input, we prove by a cosine operator/functional analytical approach that not only is the solution of the wave equation and its derivatives continuous in time, with their pointwise values in a basic energy space (in the interior of Ω), but also that a trace regularity thereof can be assigned for the solution’s time derivative in an appropriate (negative) Sobolev space. This new-found information on the solution and its traces is crucial in handling a mathematical model derived for a particular fluid/structure interaction system.     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Strong Solutions to Non-stationary Channel Flows of Heat-Conducting Viscous Incompressible Fluids with Dissipative Heating

We study an initial-boundary-value problem for time-dependent flows of heat-conducting viscous incompressible fluids in channel-like domains on a time interval (0,T). For the parabolic system with strong nonlinearities and including the artificial (the so called “do nothing”) boundary conditions, we prove the local in time existence, global uniqueness and smoothness of the solution on a time interval (0,T ^∗), where 0<T ^∗≤T.     

Sobolev regularity of the second biharmonic problem on a rectangle

Summary It is proved that for any f &esin; L²(Ω) the weak solution of the second biharmonic problem on a rectangle satisfies u&esin; H⁴(Ω). The proof uses the decomposition of the problem into two Poisson equations and a general condition for H⁴-regularity via the eigenvalues and eigenfunctions of second order elliptic operators.     

On the zero-velocity-limit solutions to the Navier–Stokes equations of compressible flow

We consider the zero-velocity stationary problem of the Navier–Stokes equations of compressible isentropic flow describing the distribution of the density ϱ of a fluid in a spatial domain Ω⊂ℝ ^N driven by a time-independent potential external force b=∇F. A sharp condition in terms of F is given for the problem to possess a unique nonnegative solution ϱ having a prescribed mass m > 0. Received: 20 October 1997     

Exponential decay bounds for nonlinear heat problems with Robin boundary conditions

We investigate the behavior of the solution of a nonlinear heat problem, when Robin conditions are prescribed on the boundary ∂Ω × (t > 0), Ω a bounded R ² domain. We determine conditions on the geometry and data sufficient to preclude the blow up of the solution and to obtain an exponential decay bound for the solution and its gradient.