Existence and uniqueness of solutions of hyperbolic equations in divergence form with various boundary conditions on various parts of the boundary

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.     

Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.     

Classical solution of the first mixed problem for a third-order hyperbolic equation with the wave operator

We study the classical solution of the boundary value problem for a third-order hyperbolic equation defined on the plane in a half-strip with the Cauchy conditions on the lower base of the domain and with the Dirichlet conditions on the lateral boundary. By the method of characteristics, in the case of two independent variables, we find the solution of the problem in closed form and prove its uniqueness.     

On a problem with nonperiodic frequent alternation of boundary conditions imposed on fast oscillating sets

A singularly perturbed eigenvalue problem for the Laplacian in a cylinder is considered. The problem is characterized by frequent nonperiodic alternation of boundary conditions imposed on narrow strips lying on the cylinder’s lateral surface. The width of the strips is an arbitrary function of a small parameter and can oscillate rapidly, with the nature of the oscillations being arbitrary. Sharp estimates are derived for the convergence rate of the eigenvalues and eigenfunctions in the problem.     

Asymptotic analysis and boundary homogenization in linear elasticity

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of ℝ³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ϵ distributed on the boundary of this domain, when the parameter ϵ goes to 0. We use epi‐convergence arguments in order to establish this asymptotic behaviour. We then specialize this general situation to the case of identical strips of size r_ϵ ϵ‐periodically distributed on the lateral surface of an axisymmetric body. We exhibit a critical size of the strips through the limit of the non‐negative quantity −1/(ϵ ln r_ϵ) and we identify the different limit problems according to the values of this limit. Copyright © 2000 John Wiley & Sons, Ltd.     

Gap opening and split band edges in waveguides coupled by a periodic system of small windows

Using an example of two coupled waveguides, we construct a periodic second-order differential operator acting in a Euclidean domain and having spectral gaps whose edges are attained strictly inside the Brillouin zone. The waveguides are modeled by the Laplacian in two infinite strips of different width that have a common interior boundary. On this common boundary, we impose the Neumann boundary condition, but cut out a periodic system of small windows, while on the remaining exterior boundary we impose the Dirichlet boundary condition. It is shown that, by varying the widths of the strips and the distance between the windows, one can control the location of the extrema of the band functions as well as the number of the open gaps. We calculate the leading terms in the asymptotics for the gap lengths and the location of the extrema.     

On the rate of convergence for perforated plates with a small interior Dirichlet zone

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type.     

On the rate of convergence for perforated plates with a small interior Dirichlet zone

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

Inverse coefficient problems for elliptic equations in a cylinder: II

We study sufficient conditions for the unique solvability of the inverse coefficient problem. We obtain various global sufficient conditions in the form of constraints on the signs of the given functions and their derivatives. As a corollary, we consider statements of inverse coefficient problems with overdetermination on the boundary, where the Dirichlet conditions are supplemented with the vanishing condition for the normal derivative on part of the boundary. We prove sufficient conditions for the existence of a solution in this case.     

The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide

We consider the heat equation in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of t−1/2. The proof employs similarity variables that lead to a non-autonomous parabolic equation in a thin strip contracting to the real line, that can be analysed on weighted Sobolev spaces in which the operators under consideration have discrete spectra. A careful analysis of its asymptotic behaviour shows that an added Dirichlet boundary condition emerges asymptotically at the switching point, breaking the real line in two half-lines, which leads asymptotically to the 1/2 gain on the spectral lower bound, and the t−1/2 gain on the decay rate in the original physical variables.This result is an adaptation to the case of strips with twisted boundary conditions of previous results by the authors on geometrically twisted Dirichlet tubes.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

First mixed problem for a nonstrictly hyperbolic equation of the third order in a bounded domain

We study the classical solution of a boundary value problem for a nonstrictly parabolic equation of the third order in a rectangular domain of two independent variables. We pose Cauchy conditions on the lower base of the domain and the Dirichlet conditions on the lateral boundary. By the method of characteristics, we obtain a closed-form analytic expression for the solution of the problem. The uniqueness of the solution is proved.     

On homogenization of solutions of the poisson equation in a perforated domain with different types of boundary conditions on different cavities

In this paper we consider boundary value problems in perforated domains with periodic structures and cavities of different scales, with the Neumann condition on some of them and mixed boundary conditions on others. We take a case when cavities with mixed boundary conditions have so called critical size (see [1]) and cavities with the Neumann conditions have the scale of the cell. In the same way other cases can be studied, when we have the Neumann and the Dirichlet boundary conditions or the Dirichlet condition and the mixed boundary condition on the boundary of cavities.There is a large literature where homogenization problems in perforated domains were studied [2];-[7];     

Dirichlet problem for the Helmholtz equation in an exterior two-dimensional domain with cuts without matching conditions at the cut endpoints

The Dirichlet problem for the Helmholtz equation in a plane exterior domain with cuts is considered for the case in which functions defined on opposite sides of the cuts in the Dirichlet boundary condition do not necessarily satisfy the matching conditions at the cut endpoints and the solution of the problem is not necessarily continuous at the endpoints of the cuts. We give a well-posed statement of the problem, prove existence and uniqueness theorems for a classical solution, derive an integral representation of the solution, and use it to study its properties. We show that the Dirichlet problem in the considered setting does not necessarily have a weak solution, although there exists a classical solution. We derive asymptotic formulas describing the behavior of the gradient of the solution at the endpoints of the cuts.     

Inverse Problems for Multidimensional Heat Equations by Measurements at a Single Point on the Boundary

In this article we recover a coefficient in a multidimensional inverse problem for a heat equation. We show that a sequence of measurements taken at the same point on the boundary but at different times is enough to determine the coefficient uniquely. We provide an identifiability algorithm for both Dirichlet and Neumann lateral boundary conditions and we examine the smoothness of the recovered coefficient.     

Asymptotic approximations of solutions to parabolic boundary value problems in thin perforated domains of rapidly varying thickness

We consider initial boundary value problems for parabolic differential equations with rapidly oscillating coefficients in thin perforated domains of rapidly varying thickness. Under certain symmetry conditions on the domain and coefficients, we construct an asymptotic expansion of a solution to the problem with homogeneous third kind conditions on the exterior boundary and the boundary of cavities. In the case of inhomogeneous Neumann conditions, we construct an asymptotic solution without symmetry assumptions and prove an asymptotic estimate in the corresponding Sobolev space. Bibliography: 27 titles. Illustrations: 1 figure.     

On a nonclassical interpretation of the dirichlet problem for a fourth-order pseudoparabolic equation

For a fourth-order pseudoparabolic equation with nonsmooth coefficients in a rectangular domain, we consider the Dirichlet problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions and the classical boundary conditions for the case in which the solution of the problem is sought in a Sobolev space.     

A problem for the Helmholtz equation outside cuts on the plane with the dirichlet condition and the oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the Helmholtz equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition with pure imaginary coefficient of the tangential derivative is posed on the other side. We prove the uniqueness of the solution. The solvability of the problem is proved for the case in which the above-mentioned pure imaginary coefficient is less than unity in absolute value. In this case, we obtain an integral representation of the solution of the problem in the form of potentials. The densities of the potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. The boundary value problem considered here generalizes the mixed Dirichlet-Neumann problem.     

Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions

The main subject of this work is to study the concept of very weak solution for the hydrostatic Stokes system with mixed boundary conditions (non-smooth Neumann conditions on the rigid surface and homogeneous Dirichlet conditions elsewhere on the boundary). In the Stokes framework, this concept has been studied by Conca [Rev. Mat. Apl. 10 (1989)] imposing non-smooth Dirichlet boundary conditions.In this paper, we introduce the dual problem that turns out to be a hydrostatic Stokes system with non-free divergence condition. First, we obtain strong regularity for this dual problem (which can be viewed as a generalisation of the regularity results for the hydrostatic Stokes system with free divergence condition obtained by Ziane [Appl. Anal. 58 (1995)]). Afterwards, we prove existence and uniqueness of very weak solution for the (primal) problem.As a consequence of this result, the existence of strong solution for the non-stationary hydrostatic Navier-Stokes equations is proved, weakening the hypothesis over the time derivative of the wind stress tensor imposed by Guillén-González, Masmoudi and Rodríguez-Bellido [Differential Integral Equations 50 (2001)].