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August  2019, 24(8): 4217-4246. doi: 10.3934/dcdsb.2019079

## Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary

 1 Dpto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain 2 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain 3 Depto. de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil 4 Depto. Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil

Dedicated to Peter Kloeden on his 70th aniversary

Received  April 2018 Revised  October 2018 Published  August 2019 Early access  April 2019

In this paper we analyze the asymptotic behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating region with reaction terms concentrated in a neighborhood of the oscillatory boundary $\theta_\varepsilon \subset\Omega_{\varepsilon }\subset \mathbb{R}^2$ when a small parameter $\varepsilon >0$ goes to zero. Our main result is concerned with the upper and lower semicontinuity of the set of solutions in $H^1$. We show that the solutions of our perturbed equation can be approximated with one defined in a fixed limit domain, which also captures the effects of reaction terms that take place in the original problem as a flux condition on the boundary of the limit domain.

Citation: José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4217-4246. doi: 10.3934/dcdsb.2019079
##### References:

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##### References:
The oscillatory domain $\Omega_ \varepsilon$ and strip $\theta_ \varepsilon$ where reactions take place
Fixed $x_1\in(0,1)$ and $\varepsilon>0$, we get a fiber of the oscillatory domain for $\varepsilon<\varepsilon_0$
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