JournalsaihpcVol. 23, No. 4pp. 499–538

Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems

  • Kai Medville

    Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
  • Michael S. Vogelius

    Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems cover
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Abstract

In this paper we study, analytically and numerically, the existence and blow up of solutions to two-dimensional boundary value problems of the form Δuλ=0\mathrm{\Delta }u_{\lambda } = 0 in Ω, uλ/n=Duλ+λf(uλ)\partial u_{\lambda }/ \partial \mathbf{n} = Du_{\lambda } + \lambda f(u_{\lambda }) on ∂Ω. We place particular emphasis on f(u)=sinh(u)=(eueu)/2f(u) = \mathrm{\sinh }(u) = {}^{(\mathrm{e}^{u}−\mathrm{e}^{−u})}/_{2}, in which case the nonlinear flux boundary condition is frequently associated with the names of Butler and Volmer.

Résumé

Dans cet article nous étudions, analytiquement et numériquement, l'existence et l'explosion de solutions de problèmes aux limites bi-dimensionnels de la forme Δuλ=0\mathrm{\Delta }u_{\lambda } = 0 dans Ω, uλ/n=Duλ+λf(uλ)\partial u_{\lambda }/ \partial \mathbf{n} = Du_{\lambda } + \lambda f(u_{\lambda }) sur ∂Ω. Nous portons une attention particulière à f(u)=sinh(u)=(eueu)/2f(u) = \mathrm{\sinh }(u) = {}^{(\mathrm{e}^{u}−\mathrm{e}^{−u})}/_{2}, situation dans laquelle la condition non linéaire sur le flux au bord est fréquemment associée aux noms de Butler et Volmer.

Cite this article

Kai Medville, Michael S. Vogelius, Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 4, pp. 499–538

DOI 10.1016/J.ANIHPC.2005.02.008