Fourier Boundary Conditions Research Articles

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.

Homogenization of 3D thick cascade junction with a random transmission zone periodic in one direction

In the paper, we deal with the homogenization problem for the Poisson equation in a singularly perturbed three-dimensional junction of a new type. This junction consists of a body and a large number of thin curvilinear cylinders, joining to body through a random transmission zone with rapidly oscillating boundary, periodic in one direction. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and with random perturbed coefficients on the boundary of the transmission zone. We prove the homogenization theorems and the convergence of the energy integrals.

Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

We study the asymptotic behavior (as e→0) of an optimal control problem in a plane thick two-level junction, which is the union of some domain and a large number 2N of thin rods with variable thickness of order \(\varepsilon =\mathcal{O}(N^{-1}).\) The thin rods are divided into two levels depending on the geometrical characteristics and on the controls given on their bases. In addition, the thin rods from each level are e-periodically alternated and the inhomogeneous perturbed Fourier boundary conditions are given on the lateral sides of the rods. Using the direct method of the calculus of variations and the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis of this problem for different kinds of controls is made as e→0.

Asymptotic analysis of a boundary-value problem in a cascade thick junction with a random transmission zone

In the article we deal with the homogenization of a boundary-value problem for the Poisson equation in a singularly perturbed two-dimensional junction of a new type. This junction consists of a body and a large number of thin rods, which join the body through the random transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin rods and with random perturbed coefficients on the boundary of the transmission zone. We prove the homogenization theorems and the convergence of the energy integrals. It is shown that there are three qualitatively different cases in the asymptotic behaviour of the solutions.

Γ-convergence approach to variational problems in perforated domains with Fourier boundary conditions

The work focuses on the Γ-convergence problem and the convergence of minimizers for a functional defined in a periodic perforated medium and combining the bulk (volume distributed) energy and the surface energy distributed on the perforation boundary. It is assumed that the mean value of surface energy at each level set of test function is equal to zero. Under natural coercivity and p-growth assumptions on the bulk energy, and the assumption that the surface energy satisfies p-growth upper bound, we show that the studied functional has a nontrivial Γ-limit and the corresponding variational problem admits homogenization.

Homogenization of a boundary value problem in a thick cascade junction

We consider a homogenization problem in a singularly perturbed two-dimensional domain of a new type that consists of a junction body and many alternating thin rods of two classes. One of the classes consists of rods of finite length, whereas the other contains rods of small length, and inhomogeneous Fourier boundary conditions (the third type boundary conditions) with perturbed coefficients are imposed on boundaries of thin rods. Homogenization theorems are proved. Bibliography: 38 titles. Illustrations: 2 figures.

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ℝ N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.

Homogenization of nonlinear parabolic problems with varying boundary conditions on varying sets

This paper reports on a study of the asymptotic behaviour of the solution of a nonlinear parabolic problem posed on a sequence of varying domains. We also consider that the solution satisfies a Neumann boundary condition on an arbitrary sequence of subsets of the boundary and a Dirichlet boundary condition on the remainder of it. Assuming that the operators do not depend on time, we show that the corrector obtained for the elliptic problem, still gives a corrector for the parabolic problem. From this result, we obtain the limit problem which is stable by homogenization and where it appears, a generalized Fourier boundary condition.

Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets

AbstractWe study the asymptotic behaviour of the solution un of a linear elliptic equation posed in a fixed domain Ω. The solution un is assumed to satisfy a Dirichlet boundary condition on Γn, where Γn is an arbitrary sequence of subsets of ∂Ω, and a Neumman boundary condition on the remainder of ∂Ω. We obtain a representation of the limit problem which is stable by homogenization and where it appears a generalized Fourier boundary condition. We also prove a corrector result. Copyright © 2007 John Wiley & Sons, Ltd.

Analysis of Direct Three-Dimensional Parabolic Panel Methods

Adherence boundary conditions for time dependent partial differential equations, via Chorin algorithm, can be reduced to a parabolic problem with Robin–Fourier boundary conditions in the three-dimensional context. In the spirit of panel methods, one establishes an integral formulation whose key point is the estimation of the potential density, introducing a kind of panel method for tangential kinematic boundary conditions. This paper discusses explicit estimations of this density in the general case of an arbitrarily shaped three-dimensional body, which leads to a fast numerical scheme. An error analysis is also provided, involving body smoothness, the Hölder exponent of the density, and whether the body presents torsion or not.

Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case

This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form ∂y + f(y) = 0. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one has exact controllability to the trajectories. Mathematics Subject Classification. 35K20, 93B05.

Null controllability of the heat equation with boundary Fourier conditions: the linear case

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂y ∂n + βy = 0. We consider distributed controls with support in a small set and nonregular coefficients β = β(x, t). For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.

ON A LUBRICATION PROBLEM WITH FOURIER AND TRESCA BOUNDARY CONDITIONS

The asymptotic behavior of a Stokes flow with Fourier boundary condition on one part on the boundary and Tresca free boundary friction condition on the other, when one dimension of the fluid domain tends to zero is studied. The strong convergence of the velocity is proved, a specific Reynolds equation is obtained, and the uniqueness of the limit velocity and pressure distributions is established.

The Stokes equations with Fourier boundary conditions on a wall with asperities

AbstractWe study the effect of the rugosity of a wall on the solution of the Stokes system complemented with Fourier boundary conditions. We consider the case of small periodic asperities of size ε. We prove that the velocity field, pressure and drag, respectively, converge to the velocity field, pressure and drag of a homogenized Stokes problem, where a different friction coefficient appears. This shows that, contrarily to the case of Dirichlet boundary conditions, rugosity is dominant here. Copyright © 2001 John Wiley & Sons, Ltd.

Effet de la rugosité sur un fluide laminaire avec conditions de Fourier

The effect of tiny asperities covering a wall on a flow governed by Stokes equations with Fourier boundary conditions is investigated. We calculate the limit flow and we give estimates of the deviations of the drag, velocity field and pressure, in terms of the size ε of the asperities. In the particular case of a plate, the limit drag is larger than the drag of the smooth wall, in contrast with the situation found for Dirichlet boundary conditions.

Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method

In this paper, we prove, following [1], existence and uniqueness of the solutions of convection-diffusion equations on an open subset of $\mathbb R^N$, with a measure as data and different boundary conditions: mixed, Neumann or Fourier. The first part is devoted to the proof of regularity results for solutions of convection-diffusion equations with these boundary conditions and data in $(W^{1,q}(\Omega))'$, when $q <N/(N-1)$. The second part transforms, thanks to a duality trick, these regularity results into existence and uniqueness results when the data are measures.

The Boundary-value Problem in Domains with Very Rapidly Oscillating Boundary

We study the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain Ωε⊂Rnwith a very rapidly oscillating locally periodic boundary. We assume that the Fourier boundary condition involving a small positive parameter ε is posed on the oscillating part of the boundary and that the (n−1)-dimensional volume of this part goes to infinity as ε→0. Under proper normalization conditions that homogenized problem is found and the estimates of the residual are obtained. Also, we construct an additional term of the asymptotics to improve the estimates of the residual. It is shown that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem.

Boundary layer tails in periodic homogenization

This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period e): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.

Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.