Marcone C. Pereira

Homogenization in a thin domain with an oscillatory boundary

Abstract In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type R ϵ = { ( x 1 , x 2 ) ∈ R 2 | x 1 ∈ ( 0 , 1 ) , 0 x 2 ϵ G ( x 1 , x 1 / ϵ ) } where the function G ( x , y ) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter ϵ.

The Neumann problem in thin domains with very highly oscillatory boundaries

Abstract In this paper, we analyze the behavior of solutions of the Neumann problem posed in a thin domain of the type R ϵ = { ( x 1 , x 2 ) ∈ R 2 ∣ x 1 ∈ ( 0 , 1 ) , − ϵ b ( x 1 ) x 2 ϵ G ( x 1 , x 1 / ϵ α ) } with α > 1 and ϵ > 0 , defined by smooth functions b ( x ) and G ( x , y ) , where the function G is supposed to be l ( x ) -periodic in the second variable y . The condition α > 1 implies that the upper boundary of this thin domain presents a very high oscillatory behavior. Indeed, we have that the order of its oscillations is larger than the order of the amplitude and height of R ϵ given by the small parameter ϵ . We also consider more general and complicated geometries for thin domains which are not given as the graph of certain smooth functions, but rather more comb-like domains.

Elliptic problems in thin domains with highly oscillating boundaries

We study the Laplace operator with Neumann boundary conditions in a 2-dimensional thin domain with a higly oscillating boundary. We obtain the correct limit problem for the case where the boundary is the graph of the oscillating function ϵGϵ(x) where Gϵ(x) = a(x) + b(x)g(x/ϵ) with g periodic and a and b not necessarily constant.

Remarks on the p-Laplacian on thin domains

The limiting behavior of solutions of quasilinear elliptic equations on thin domains is investigated. As we will see the boundary conditions play an important role. If one considers homogeneous Dirichlet boundary conditions, the sequence of solutions will converge to the null function, whereas, if one considers Neumann boundary conditions, there is a nontrivial equation which determines the limiting behavior.

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady-state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a e-neighborhood of a portion Γ of the boundary. We assume that this e-neighborhood shrinks to Γ as the small parameter e goes to zero. Also, we suppose the upper boundary of this e-strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright

A generic property for the eigenfunctions of the Laplacian

In this work we show that, generically in the set of

Generic Simplicity for the Solutions of a Nonlinear Plate Equation

\mathcal{C}^2

Nonlocal evolution problems in thin domains

bounded regions of

Continuity of attractors for a family of \({\mathcal {C}}^1\) perturbations of the square

\mathbb R^n

A Parallel PCA Neural Network Approach for Feature Extraction

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