Heterogeneous time-harmonic Maxwell equations in bidimensional domains

This note deals with the low-frequency time-harmonic Maxwell equations for a heterogeneous media in bidimensional bounded domains. We propose a three step method to solve this problem. First, we construct an extension of the boundary data solving a scalar Neumann problem for the Laplace operator. Second, we solve a problem in the conductor with an unusual boundary condition of nonlocal type. Third, we solve a boundary value problem in the insulator using the solution calculated in the conductor. Also, this third problem can be reduced to a Neumann problem for the Laplace operator.     

Solvability of a boundary value problem for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid

A boundary value problem is studied for a stationary model of the magnetic hydrodynamics of a viscous heat-conducting fluid under nonhomogeneous boundary conditions on the velocity, electromagnetic field, and temperature. The model consists of the Navier-Stokes equations, the Maxwell equations, the generalized Ohm law, and the convection-diffusion equation for the temperature which are connected nonlinearly with each other. Sufficient conditions on the initial data are established that guarantee the global solvability of the problem under consideration and the local uniqueness of its solution. The properties are studied of the linear operator obtained by linearizing the operator of the original boundary value problem.     

Inverse boundary spectral problem for non–selfadjoint maxwells–s equations with incomplete data

The inverse boundary spectral problem for selfadjoint Maxwell–s equations is to reconstruct unknown coefficient functions in Maxwell– equations from the knowledge of the boundary spectral data, i.e. fromt eh eigenvalues and the boudnary value of the eigenfunctions. Since the spectrum of non–selfadjoint Maxwell–s operator consists of normal eigenvalues and an interval, the complete boundary spectral data can be defind only in a very complicated way. In this article we show that the coefficients can be reconstructed from incomplete data, that is, from the large eigenvalues and the boundary values of the generalized eigenfunctions. Particularly, we do not need the nfinit–dimensional data corresponding to the non–discrete spectrum.     

The energy method for constructing time-harmonic solutions to the maxwell equations

We propose some minimum principle for an energy functional in an elliptic boundary value problem that arises in constructing time-harmonic solutions to the Maxwell equations. We suggest the potentials other than the vector and scalar potentials, used in the mathematical modeling of electromagnetic fields since the operators of traditional problems are not sign definite, which complicates constructions of iterative solution methods. We consider the problem in a parallelepiped whose boundary is ideally conducting. For nonresonant frequencies we prove that the operator of the boundary value problem is positive definite, propose a minimum principle for a quadratic energy functional, and prove the existence and uniqueness of generalized solutions.     

A Class of Evolutionary Problems with an Application to Acoustic Waves with Impedance Type Boundary Conditions.

A well-posedness result for a time-shift invariant class of evolutionary operator equations is considered and exemplified by an application to an impedance type initial boundary value problem for the system of linear acoustics. The problem class allows for memory effects in the domain as well as on the domain boundary. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account. To cite this article: A. Buffa et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the Fredholm property of the electric field equation in the vector diffraction problem for a partially screened solid

We consider a vector problem of diffraction of an electromagnetic wave on a partially screened anisotropic inhomogeneous dielectric body. The boundary conditions and the matching conditions are posed on the boundary of the inhomogeneity domain, and under passage through it, the medium parameters have jump changes. A boundary value problem for the system of Maxwell equations in unbounded space is studied in a semiclassical statement and is reduced to a system of integro-differential equations on the body domain and the screen surfaces. We show that the quadratic form of the problem operator is coercive and the operator itself is Fredholm with zero index.     

Inverse final observation problems for Maxwell’s equations in the quasi-stationary magnetic approximation and stable sequential Lagrange principles for their solving

An initial–boundary value problem for Maxwell’s equations in the quasi-stationary magnetic approximation is investigated. Special gauge conditions are presented that make it possible to state the problem of independently determining the vector magnetic potential. The well-posedness of the problem is proved under general conditions on the coefficients. For quasi-stationary Maxwell equations, final observation problems formulated in terms of the vector magnetic potential are considered. They are treated as convex programming problems in a Hilbert space with an operator equality constraint. Stable sequential Lagrange principles are stated in the form of theorems on the existence of a minimizing approximate solution of the optimization problems under consideration. The possibility of applying algorithms of dual regularization and iterative dual regularization with a stopping rule is justified in the case of a finite observation error.     

Initial value problems for creeping flow of Maxwell fluids

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data.We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.     

On a model of electromagnetic field propagation in ferroelectric media

The Maxwell system in an anisotropic, inhomogeneous medium with non-linear memory effect produced by a Maxwell type system for the polarization is investigated under low regularity assumptions on data and domain. The particular form of memory in the system is motivated by a model for electromagnetic wave propagation in ferromagnetic materials suggested by Greenberg, MacCamy and Coffman [J.M. Greenberg, R.C. MacCamy, C.V. Coffman, On the long-time behavior of ferroelectric systems, Phys. D 134 (1999) 362-383]. To avoid unnecessary regularity requirements the problem is approached as a system of space-time operator equation in the framework of extrapolation spaces (Sobolev lattices), a theoretical framework developed in [R. Picard, Evolution equations as space-time operator equations, Math. Anal. Appl. 173 (2) (1993) 436-458; R. Picard, Evolution equations as operator equations in lattices of Hilbert spaces, Glasnik Mat. 35 (2000) 111-136]. A solution theory for a large class of ferromagnetic materials confined to an arbitrary open set (with suitably generalized boundary conditions) is obtained.     

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.     

A Variational Approach for the Mixed Problem in the Elastostatics of Bodies with Dipolar Structure

In this study, we address the mixed initial boundary value problem in the elastostatics of dipolar bodies. Using the equilibrium equations, we build the operator of dipolar elasticity and prove that this operator is positively defined even in the general case of an elastic inhomogeneous and anisotropic dipolar solid. This helps us to prove the existence of a generalized solution for first boundary value problem and also the uniqueness of the solution. Moreover, relying on this property of the operator of dipolar elasticity to be positively defined, we can apply the known variational method proposed by Mikhlin.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

On the unique existence of the classical solution to the problem of electromagnetic wave diffraction by an inhomogeneous lossless dielectric body

A vector problem of electromagnetic wave diffraction by an inhomogeneous volumetric body is considered in the classical formulation. The uniqueness theorem for the solution to the boundary value problem for the system of Maxwell’s equations is proven in the case when the permittivity is real and varies jumpwise on the boundary of the body. A vector integro-differential equation for the electric field is considered. It is shown that the operator of the equation is continuously invertible in the space of square-summable vector functions.     

Boundary element approximation for Maxwell's eigenvalue problem

We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of the coincidence set of a solution to a parabolic variational inequality with an obstacle

In this paper we propose a new technique for the stability analysis of the coincidence set of a solution to a parabolic variational inequality with an obstacle inside the domain. It is based on the reformulation of the initial inequality in the form of a parabolic initial boundary value problem with an exact penalty operator.     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.     

Investigation of electromagnetic wave diffraction from an inhomogeneous partially shielded solid

Diffraction of electromagnetic wave from a partially shielded inhomogeneous dielectric is considered. The original boundary value problem for Maxwell’s equations is shown to have at most one quasi-classical solution. The problem is reduced to a system of integro-differential equations on the solid and the screens. The matrix integro-differential operator is treated in Sobolev spaces and is shown to be a continuously invertible operator. As a result, convergence of the Galerkin method is proved in the chosen functional spaces.     

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.