Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

Teresa D’Aprile

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 2, page 219-259
  • ISSN: 0391-173X

Abstract

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We consider the problemwhere Ω ⊂ ℝ 3 is a smooth and bounded domain, ε , γ 1 , γ 2 > 0 , v , V : Ω → ℝ , f : ℝ → ℝ . We prove that this system has aleast-energy solution v ε which develops, as ε → 0 + , a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches themost curvedpart of ∂ Ω ,i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solutions in powers of ε up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in Ω .

How to cite

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D’Aprile, Teresa. "Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.2 (2006): 219-259. <http://eudml.org/doc/243796>.

@article{D2006,
abstract = {We consider the problemwhere $\Omega \subset \mathbb \{R\}^3$ is a smooth and bounded domain, $\varepsilon ,\,\gamma _1,\,\gamma _2&gt;0,$$v,\,V:\Omega \rightarrow \mathbb \{R\}$, $f:\mathbb \{R\}\rightarrow \mathbb \{R\}$. We prove that this system has aleast-energy solution$v_\varepsilon $ which develops, as $\varepsilon \rightarrow 0^+$, a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches themost curvedpart of $\partial \Omega $,i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solutions in powers of $\varepsilon $ up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in $\Omega $.},
author = {D’Aprile, Teresa},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {219-259},
publisher = {Scuola Normale Superiore, Pisa},
title = {Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem},
url = {http://eudml.org/doc/243796},
volume = {5},
year = {2006},
}

TY - JOUR
AU - D’Aprile, Teresa
TI - Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 2
SP - 219
EP - 259
AB - We consider the problemwhere $\Omega \subset \mathbb {R}^3$ is a smooth and bounded domain, $\varepsilon ,\,\gamma _1,\,\gamma _2&gt;0,$$v,\,V:\Omega \rightarrow \mathbb {R}$, $f:\mathbb {R}\rightarrow \mathbb {R}$. We prove that this system has aleast-energy solution$v_\varepsilon $ which develops, as $\varepsilon \rightarrow 0^+$, a single spike layer located near the boundary, in striking contrast with the result in [37] for the single Schrödinger equation. Moreover the unique peak approaches themost curvedpart of $\partial \Omega $,i.e., where the boundary mean curvature assumes its maximum. Thus this elliptic system, even though it is a Dirichlet problem, acts more like a Neumann problem for the single-equation case. The technique employed is based on the so-called energy method, which consists in the derivation of an asymptotic expansion for the energy of the solutions in powers of $\varepsilon $ up to sixth order; from the analysis of the main terms of the energy expansion we derive the location of the peak in $\Omega $.
LA - eng
UR - http://eudml.org/doc/243796
ER -

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