Boundary-transmission Problems Research Articles

Stabilization of string–beam–string transmission systems with localized frictional damping and delay

In this paper, we consider two string–beam–string boundary transmission problems with localized frictional damping and delay. The first transmission problem is that the frictional damping and delay are in the left string equation and the right string equation is thermoelastic. The second transmission problem is that the damping and the delay are both only acting on the middle beam equation. Using the semigroup theory, we get the existence and the uniqueness of solution under suitable assumptions on the weight of the damping and the weight of the delay. Furthermore, we prove that the first transmission problem is exponentially stable and the second transmission problem is polynomially stable using the frequency domain method and the multiplier techniques.

Mathematical problems of dynamical interaction of fluids and multiferroic solids

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region is immersed in an inviscid fluid occupying an unbounded domain . In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Stationary states to a free boundary transmission problem for an electrostatically actuated plate

A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler–Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.

A Wave Diffraction Problem with Higher Order Impedance Boundary Conditions

In this paper, we consider an impedance boundary transmission problem for the Helmholtz equation originated by a problem of wave diffraction by an infinite strip with higher order imperfect boundary conditions. Operator theoretical methods and relations between operators are built to deal with the problem and, as consequence, a transparent interpretation of the problem in an operator theory framework are associated to the problem. In particular, different types of operator relations are exhibited for different types of operators acting between Lebesgue and Sobolev spaces on a finite interval and the positive half-line. All this has consequences in the understanding of the structure of this type of problems. At the end, we describe when the operators associated with the problem enjoy the Fredholm property in terms of the initial space order parameters.

Mixed Boundary Transmission Problems for the Linear Theory of Elasticity

Mixed Boundary Transmission Problems for the Linear Theory of Elasticity

Mixed boundary–transmission problems for composite layered elastic structures

We investigate mixed type boundary–transmission problems of the generalized thermo‐electro‐magneto‐elasticity theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary–transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide.

Shape Derivative of the Dirichlet Energy for a Transmission Problem

For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-empty intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator.

Heterogeneous Dielectric Properties in Models for Microelectromechanical Systems

An idealized electrostatically actuated microelectromechanical system involving an elastic plate with a heterogeneous dielectric material is considered. Starting from the electrostatic and mechanical energies, the governing evolution equations for the electrostatic potential and the plate deflection are derived from the corresponding energy balance. This leads to a free boundary transmission problem due to a jump of the dielectric permittivity across the interface separating the elastic plate and free space. Reduced models retaining the influence of the heterogeneity of the elastic plate under suitable assumptions are obtained when either the elastic's plate thickness or the aspect ratio of the device vanishes.

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body ( Ω + ) is embedded in an unbounded fluid domain ( Ω − ). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω + , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω − , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.

Mathematical problems of interaction of different dimensional physical fields

In the paper we consider mathematical problem of interaction of different dimensional physical fields for complex composite structures. We investigate the mixed transmission problem arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body (Ω+) is embedded in an unbounded fluid (Ω−). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω+ we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω− we have a scalar Helmholtz equation describing an acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations the uniqueness and existence theorems are proved in Sobolev-Slobodetski spaces.

The two straight line approach for periodic diffraction boundary-value problems

Boundary-transmission problems of first order for the Helmholtz equation are considered within the context of wave diffraction by a periodic strip grating and formulated as convolution type operators acting on a Bessel potential periodic space setting. Two boundary-value problems are studied for an arbitrary geometry of the grating: the oblique derivative and the classic Neumann boundary-value problems. The convolution type operators on the grating which correspond to the given boundary-transmission problems are associated with Toeplitz operators acting on spaces of matrix functions defined on composed contours. A Fredholm theory for periodic boundary-value problems of first order is established independently of the grating period and the Fredholm indices for the oblique derivative and the classic Neumann boundary-value problems are given.

An operator approach for an oblique derivative boundary‐transmission problem

AbstractWe consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.

Some initial boundary problems in electrodynamics for canonical domains in quaternions

The initial boundary-transmission problems for electromagnetic fields in homogeneous and anisotropic media for canonical semi-infinite domains, like halfspaces, wedges and the exterior of half- and quarter-plane obstacles are formulated with the use of complex quaternions. The time-harmonic case was studied by A. Passow in his Darmstadt thesis 1998 in which he treated also the case of an homogeneous and isotropic layer in free space and above an ideally conducting plane. For thin layers and free space on the top a series of generalized vectorial Leontovich boundary value conditions were deduced and systems of Wiener-Hopf pseudo-differential equations for the tangential components of the electric and magnetic field vectors and their jumps across the screens were formulated as equivalent unknowns in certain anisotropic boundary Sobolev spaces. Now these results may be formulated with alternating differential forms in Lorentz spaces or with complex quaternions.

The double layer potential method for a boundary transmission problem for the Laplace operator in an infinite wedge

AbstractThis paper is concerned with the solution of a boundary transmission problem in an infinite wedge. We treat this problem by a boundary integral method using Green's contact function for two half‐spaces. The integral operators are studied via a harmonic analysis approach which goes back to a paper of Fabes et al. We improve their results studying the Fourier symbol of the associated integral operators on the half‐plane. This leads to invertibility criteria for the boundary integral operators.

Sommerfeld Half‐plane Problems with Higher Order Boundary Conditions

AbstractSommerfeld‐type diffraction problems for a half‐plane with arbitrary n‐th order generalized impedance boundary conditions arc examined in a Sobolev space setting. The corresponding boundary‐transmission problems for the two dimensional Helmholtz equation are shown to be well‐posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener‐Hopf type in [L2+ (IR)]2. Formulas for the solutions as well as the so‐called edge conditions arc obtained for any n, by explicit canonical generalized factorization of the presymbols of the associated Wiener‐Hopf operators.

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

AbstractThis paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.

On a non-stationary free boundary transmission problem with continuous extraction and convection, arising in industrial processes

On a non-stationary free boundary transmission problem with continuous extraction and convection, arising in industrial processes