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

Teresa D’Aprile

Displaying similar documents to “Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem”

On singular perturbation problems with Robin boundary condition

Henri Berestycki, Juncheng Wei (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the following singularly perturbed elliptic problem $\begin{matrix} ϵ^{2} Δ u - u + f (u) & = 0, u > 0 in Ω, \\ ϵ \frac{\partial u}{\partial ν} + λ u & = 0 on \partial Ω, \end{matrix}$ where $f$ satisfies some growth conditions, $0 \leq λ \leq + \infty$ , and $Ω \subset ℝ^{N}$ ( $N > 1$ ) is a smooth and bounded domain. The cases $λ = 0$ (Neumann problem) and $λ = + \infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant $λ_{*} > 1$ such that, as $ϵ \to 0$ , the least energy solution has a spike near the boundary if $λ \leq λ_{*}$ , and has an interior spike near the innermost part of the domain if $λ > λ_{*}$ . Central...

Ginzburg-Landau vortices : the static model

Tristan Rivière (1999-2000)

Séminaire Bourbaki

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Non-vanishing of class group $L$ -functions at the central point

Valentin Blomer (2004)

Annales de l’institut Fourier

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Let $K = ℚ (\sqrt{- D})$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c > 0$ such that for sufficiently large $D$ at least $c \cdot h \prod_{p ∣ D} (1 - p^{- 1})$ of the $h$ distinct $L$ -functions $L_{K} (s, χ)$ do not vanish at the central point $s = 1 / 2$ .

Improved estimates for the Ginzburg-Landau equation : the elliptic case

Fabrice Bethuel, Giandomenico Orlandi, Didier Smets (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $G L$ -energy $E_{ε}$ and the parameter $ε$ . These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

A free boundary value problem in potential theory

Guido Stampacchia, D. Kinderlehrer (1975)

Annales de l'institut Fourier

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This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain $Ω$ and a function $u$ defined in $Ω$ satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of $Ω$ . The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists...

The proof of Minkowski’s conjecture concerning the critical determinant of the region ${| x |}^{p} + {| y |}^{p} < 1$ for p ≥ 6

A. Malyshev, A. Voronetsky (1975)

Acta Arithmetica

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Displaying similar documents to “Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem”

On singular perturbation problems with Robin boundary condition

Ginzburg-Landau vortices : the static model

Non-vanishing of class group L -functions at the central point

Improved estimates for the Ginzburg-Landau equation : the elliptic case

A free boundary value problem in potential theory

The proof of Minkowski’s conjecture concerning the critical determinant of the region | x | p + | y | p < 1 for p ≥ 6

Non-vanishing of class group $L$ -functions at the central point

The proof of Minkowski’s conjecture concerning the critical determinant of the region ${| x |}^{p} + {| y |}^{p} < 1$ for p ≥ 6