Higher order energy expansions for some singularly perturbed Neumann problems

Juncheng Wei; Matthias Winter

doi:10.1016/S1631-073X(03)00269-3

Partial Differential Equations

Higher order energy expansions for some singularly perturbed Neumann problems
[Développement asymptotique de l'énergie des solutions des problèmes de perturbations singulières]

Présenté par : Haïm Brézis

Juncheng Wei ¹ ; Matthias Winter ²

¹ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
² Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 37-42.

Résumé
Abstract

Nous étudions le problème suivant de perturbations singulières :

ϵ^{2} Δ u - u + u^{p} = 0 dans Ω, u > 0 dans Ω et \frac{\partial u}{\partial ν} = 0 sur \partial Ω,

où

Ω

est un domaine ouvert dans

ℝ^{N}

, ε>0 est une constante petite et p est un exposant souscritique. L'énergie s'écrit alors

J_{ϵ} [u] : = \int_{Ω} (\frac{ϵ^{2}}{2} {| \nabla u |}^{2} + \frac{1}{2} u^{2} - \frac{1}{p + 1} u^{p + 1}) d x

, où

u \in H^{1} (Ω)

. Ni et Takagi montrent que pour une solution u_ε avec une pic sur la frontière du domaine, on a le développement asymptotique suivant :

J_{ϵ} [u_{ϵ}] = ϵ^{N} [\frac{1}{2} I [w] - c_{1} ϵ H (P_{ϵ}) + o (ϵ)],

où c₁>0 est une constante générique, P_ε est le point unique de maximum local de u_ε et H(P_ε) est la fonction de la courbure moyenne sur la frontière. On établit que :

J_{ϵ} [u_{ϵ}] = ϵ^{N} [\frac{1}{2} I [w] - c_{1} ϵ H (P_{ϵ}) + ϵ^{2} [c_{2} {(H (P_{ϵ}))}^{2} + c_{3} R (P_{ϵ})] + o (ϵ^{2})],

où c₂, c₃ sont les constantes génériques et R(P_ε) est la courbure scalaire de Ricci en P_ε. En particulier c₃>0. Nous présentons des applications de ce développement asymptotique.

We consider the following singularly perturbed semilinear elliptic problem:

ϵ^{2} Δ u - u + u^{p} = 0 in Ω, u > 0 in Ω and \frac{\partial u}{\partial ν} = 0 on \partial Ω,

where

Ω

is a bounded smooth domain in

ℝ^{N}

, ε>0 is a small constant and p is a subcritical exponent. Let

J_{ϵ} [u] : = \int_{Ω} (\frac{ϵ^{2}}{2} {| \nabla u |}^{2} + \frac{1}{2} u^{2} - \frac{1}{p + 1} u^{p + 1}) d x

be its energy functional, where

u \in H^{1} (Ω)

. Ni and Takagi proved that for a single boundary spike solution u_ε, the following asymptotic expansion holds

J_{ϵ} [u_{ϵ}] = ϵ^{N} [\frac{1}{2} I [w] - c_{1} ϵ H (P_{ϵ}) + o (ϵ)],

where c₁>0 is a generic constant, P_ε is the unique local maximum point of u_ε and H(P_ε) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of J_ε[u_ε]:

J_{ϵ} [u_{ϵ}] = ϵ^{N} [\frac{1}{2} I [w] - c_{1} ϵ H (P_{ϵ}) + ϵ^{2} [c_{2} {(H (P_{ϵ}))}^{2} + c_{3} R (P_{ϵ})] + o (ϵ^{2})],

where c₂, c₃ are generic constants and R(P_ε) is the Ricci scalar curvature at P_ε. In particular c₃>0. Applications of this expansion will be given.

Reçu le : 2003-05-13
Accepté le : 2003-05-13
Publié le : 2003-06-26

DOI : 10.1016/S1631-073X(03)00269-3

Affiliations des auteurs :

Juncheng Wei ¹ ; Matthias Winter ²

¹ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
² Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

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     title = {Higher order energy expansions for some singularly perturbed {Neumann} problems},
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Juncheng Wei; Matthias Winter. Higher order energy expansions for some singularly perturbed Neumann problems. Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 37-42. doi : 10.1016/S1631-073X(03)00269-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00269-3/

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