A nonlinear elliptic problem with terms concentrating in the boundary |
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| Authors: | Gleiciane S Aragão Antônio L Pereira Marcone C Pereira |
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| Institution: | 1. Universidade Federal de S?o Paulo, , Diadema, Brazil;2. Instituto de Matemática e Estatística, Universidade de S?o Paulo, , S?o Paulo, Brazil;3. Escola de Artes, Ciências e Humanidades, Universidade de S?o Paulo, , S?o Paulo, BrazilGleiciane S. Arag?o was partially supported by FAPESP 2010/51829‐7, Brazil. Ant?nio L. Pereira was partially supported by CNPq 308696/2006‐9, FAPESP 2008/55516‐3, Brazil. Marcone C. Pereira was partially supported by CNPq 305210/2008‐4, FAPESP 2008/53094‐4, 2010/18790‐0 and 2011/08929‐3, Brazil. |
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| Abstract: | 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 ε‐neighborhood of a portion Γ of the boundary. We assume that this ε‐neighborhood shrinks to Γ as the small parameter ε goes to zero. Also, we suppose the upper boundary of this ε‐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 © 2012 John Wiley & Sons, Ltd. |
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| Keywords: | semilinear elliptic equations nonlinear boundary value problems singular elliptic equations upper semicontinuity concentrating terms oscillatory behavior |
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