An example in the gradient theory of phase transitions

Camillo De Lellis

Displaying similar documents to “An example in the gradient theory of phase transitions”

Asymmetric heteroclinic double layers

Michelle Schatzman (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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Let $W$ be a non-negative function of class $C^{3}$ from $ℝ^{2}$ to $ℝ$ , which vanishes exactly at two points $𝐚$ and $𝐛$ . Let $S^{1} (𝐚, 𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $- \infty$ and to $𝐛$ at $+ \infty$ and whose one dimensional energy $E_{1} (v) = \int_{ℝ} [W (v) + | v^{'} |^{2} / 2] d x$ is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_{1}$ over $S^{1} (𝐚, 𝐛)$ . Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator...

Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^{1} (Ω)$

M. F. Betta, A. Mercaldo, F. Murat, M. M. Porzio (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is $\{\begin{matrix} - div (a (x) (1 + {| \nabla u |}^{2})^{\frac{p - 2}{2}} {\nabla u) + b (x) (1 + | \nabla u |}^{2})^{\frac{λ}{2}} = f & in Ω, \\ u = 0 & on \partial Ω, \end{matrix}$ where $Ω$ is a bounded open subset of $ℝ^{N}$ , $N \geq 2$ , $2 - 1 / N < p < N$ , $a$ belongs to $L^{\infty} (Ω)$ , $a (x) \geq α_{0} > 0$ , $f$ is a function in $L^{1} (Ω)$ , $b$ is a function in $L^{r} (Ω)$ and $0 \leq λ < λ^{*} (N, p, r),$ for some $r$ and $λ^{*} (N, p, r)$ .

On a model of rotating superfluids

Sylvia Serfaty (2001)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider an energy-functional describing rotating superfluids at a rotating velocity $ω$ , and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical $ω$ above which energy-minimizers have vortices, evaluations of the minimal energy as a function of $ω$ , and the derivation of a limiting free-boundary problem.

Impact of the variations of the mixing length in a first order turbulent closure system

Françoise Brossier, Roger Lewandowski (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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This paper is devoted to the study of a turbulent circulation model. Equations are derived from the “Navier-Stokes turbulent kinetic energy” system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity $ν_{t}$ . The mixing length $ℓ$ acts as a parameter which controls the turbulent part in $ν_{t}$ . The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of $ℓ$ and...

A spectral study of an infinite axisymmetric elastic layer

Lahcène Chorfi (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators $A_{n}$ , $n \in ℕ$ , in a suitable Hilbert space. We show that the essential spectrum of $A_{n}$ is an interval of type $[γ, + \infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

Rotating drops in a vessel

G. Congedo, M. Emmer, E. H. A. Gonzalez (1983)

Rendiconti del Seminario Matematico della Università di Padova

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The car with $N$ trailers : characterisation of the singular configurations

Frédéric Jean (1996)

ESAIM: Control, Optimisation and Calculus of Variations

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On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations

Eliane Bécache, Patrick Joly (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model....