Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure

In this paper we are concerned with convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain exhibiting highly oscillatory behavior in part of its boundary. We deal with the resonant case in which the height, amplitude and period of the oscillations are all of the same order which is given by a small parameter $\epsilon > 0$. Applying an appropriate corrector approach we get strong convergence when we replace the original solutions by a kind of first-order expansion through the Multiple-Scale Method.Comment: to appear in Quarterly of Applied Mathematic

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 $\epsilon$-neighborhood of a portion $\Gamma$ of the boundary. We assume that this $\epsilon$-neighborhood shrinks to $\Gamma$ as the small parameter $\epsilon$ goes to zero. Also, we suppose the upper boundary of this $\epsilon$-strip presents a highly oscillatory behavior. Our main goal here is 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 $\Gamma$, which depends on the oscillating neighborhood

Time-scale analysis non-local diffusion systems, applied to disease models

The objective of the present paper is to use the well known Ross-Macdonald models as a prototype, incorporating spatial movements, identifying different times scales and proving a singular perturbation result using a system of local and non-local diffusion. This results can be applied to the prototype model, where the vector has a fast dynamics, local in space, and the host has a slow dynamics, non-local in space

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with ones of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem

Error estimates for a Neumann problem in highly oscillating thin domains

In this work we analyze convergence of solutions for the Laplace operator with Neumann boundary conditions in a two-dimensional highly oscillating domain which degenerates into a segment (thin domains) of the real line. We consider the case where the height of the thin domain, amplitude and period of the oscillations are all of the same order, given by a small parameter $\epsilon$. We investigate strong convergence properties of the solutions using an appropriate corrector approach. We also give error estimates when we replace the original solutions for the second-order expansion through the Multiple-Scale Method

Nonlocal and nonlinear evolution equations in perforated domains

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form $u_t(x,t) = \int J(x-y) u(y,t) \, dy - h_\epsilon(x) u(x,t) + f(x,u(x,t))$ with $x$ in a perturbed domain $\Omega^\epsilon \subset \Omega$ which is thought as a fixed set $\Omega$ from where we remove a subset $A^\epsilon$ called the holes. We choose an appropriated families of functions $h_\epsilon \in L^\infty$ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside $\Omega$. Moreover, we take $J$ as a non-singular kernel and $f$ as a nonlocal nonlinearity. % Under the assumption that the characteristic functions of $\Omega^\epsilon$ have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation