T. A. Shaposhnikova

Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.

On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems

Abstract. Let ue be the solution of the Poisson equation in a domain periodically perforated along a manifold γ = Ω ∩ {x1 = 0}, with a nonlinear Robin type boundary condition on the perforations (the flux here being O(e−κ)σ(x,ue)), and with a Dirichlet condition on ∂Ω. Ω is a domain of R with n 3, the small parameter e, that we shall make to go to zero, denotes the period, and the size of each cavity is O(e) with α 1. The function σ involving the nonlinear process is a C1(Ω × R) function and the parameter κ ∈ R. Depending on the values of α and κ, the effective equations on γ are obtained; we provide a critical relation between both parameters which implies a different average of the process on γ ranging from linear to nonlinear. For each fixed κ a critical size of the cavities which depends on n is found. As e → 0, we show the convergence of ue in the weak topology of H1 and construct correctors which provide estimates for convergence rates of solutions. All this allows us to derive convergence for the eigenelements of the associated spectral problems in the case of σ a linear function.

Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

In this article, we consider variational inequalities arising, e.g., in modelling diffusion of substances in porous media. We assume that the media fills a domain Ωϵ of ℝ n with n ≥ 3. We study the case where the number of cavities is large and they are periodically distributed along a (n − 1)-dimensional manifold. ϵ is the period while ϵα is the size of each cavity with α ≥ 1; ϵ is a parameter that converges towards zero. Moreover, we also assume that the nonlinear process involves a large parameter ϵ−κ with κ = (α − 1)(n − 1). Passing to the scale limit, and depending on the value of α, the effective equation or variational inequality is obtained. In particular, we find a critical size of the cavities when α = κ = (n − 1)/(n − 2). We also construct correctors which improve convergence for α ≥ (n − 1)/(n − 2).

Averaging of boundary-value problem in domain perforated along (n − 1)-dimensional manifold with nonlinear third type boundary conditions on the boundary of cavities

The research considers the asymptotic behavior of solutions uɛ of the Poisson equation in a domain ɛ-periodically perforated along manifold γ = ω ∩ {x1 = 0} ≠ Ø with a nonlinear third type boundary condition ∂vuɛ + ɛ−ασ(x, uɛ) = 0 on the boundary of the cavities. It is supposed that the perforations are balls of radius C0ɛα, C0 > 0, α = n − 1 / n − 2, n ≥ 3, periodically distributed along the manifold γ with period ɛ > 0. It has been shown that as ɛ → 0 the microscopic solutions can be approximated by the solution of an effective problem which contains in a transmission conditions a new nonlinear term representing the macroscopic contribution of the processes on the boundary of the microscopic cavities. This effect was first noticed in [1] where the similar problem was investigated for n = 3 and for the case where Ω is a domain periodically perforated over the whole volume. This paper provides a new method for the proof of the convergence of the solutions {uɛ} to the solution of the effective problem is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and more accurate results are obtained with respect to the energy norm proved via a corrector term. Note that this approach can be generalized to achieve results for perforations of more complex geometry.

Homogenization of variational inequalities of Signorini type for the p-Laplacian in perforated domains when p ∈ (1, 2)

The asymptotic behavior, as ε → 0, of the solution uε to a variational inequality with nonlinear constraints for the p-Laplacian in an ε-periodically perforated domain when p ∈ (1, 2) is studied.

Boundary homogenization in perforated domains for adsorption problems with an advection term

We consider a model for the spreading of a substance through an incompressible fluid in a perforated domain , with . The fluid flows in a domain containing a periodical set of perforations () placed along an inner surface . The size of the perforations is much smaller than the size of the characteristic period . An adsorption phenomena can occur on the boundaries of the perforations, where we assume a strongly nonlinear adsorption law with a large adsorption parameter. An advection term appears in the partial differential equation. We obtain the homogenized model which also involves a nonlinear transmission condition for the normal derivative on . The ‘strange term’ arising in this transmission condition is a nonlinear function implicitly defined by a functional equation. We deal with critical relations both for the size of perforations and the adsorption parameter while we use the energy method for variational inequalities to show the convergence.

Homogenization for the p-Laplacian in an n-dimensional domain perforated by very thin cavities with a nonlinear boundary condition on their Boundary in the case p = n

We investigate the asymptotic behavior, as ε → 0, of the solution uε to the boundary value problem for the equation −Δpuε = f in a domain Ωε ⊂ ℝn perforated by very thin arbitrarily shaped cavities separated by an O(ε) distance in the case of p = n ≥ 3 with a nonlinear third boundary condition of the form

Homogenization of the p-Laplacian with nonlinear boundary condition on critical size particles: Identifying the strange term for the some non smooth and multivalued operators

\partial _{v_p } u_\varepsilon \equiv \left| {\nabla u_\varepsilon } \right|^{n - 2} \left( {\nabla u_\varepsilon ,v} \right) = - \beta ^{n - 1} \left( \varepsilon \right)\sigma \left( {x,u_\varepsilon } \right)