Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method

Giovany M. Figueiredo; João R. Santos

Displaying similar documents to “Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method”

Relaxation in BV of integrals with superlinear growth

Parth Soneji (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We study properties of the functional $\begin{matrix} ℱ_{loc} (u, Ω) : = inf_{(u_{j})} \{\underset{j \to \infty}{lim inf} \int_{Ω} f (\nabla u_{j}) x |\begin{matrix} (u_{j}) \subset W_{loc}^{1, r} (Ω,) \\ u_{j} u in (Ω,) \end{matrix}\}, \end{matrix}$ F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;R^N), and f:R^{N × n} → R is continuous and satisfies 0 ≤ f(ξ)...

Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

Luca Mugnai (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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We establish some new results about the -limit, with respect to the -topology, of two different (but related) phase-field approximations ${ℰ_{}}, {{\tilde{ℰ}}_{}}$ ℰ ε ε , x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize the-limit as → 0 of ℰ, and show that in general the -limits of ℰand $\tilde{ℰ}$ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the-limit...

Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

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Let $𝔽_{}^{}$ F be a filtration andbe a random time. Let $𝔾_{}^{}$ G be the progressive enlargement of $𝔽_{}^{}$ F with. We study the following formula, called the optional splitting formula: For any $𝔾_{}^{}$ G-optional process, there exists an $𝔽_{}^{}$ F-optional process and a function defined on [0∞] × (ℝ × ) being $ℬ [0, \infty] \otimes 𝒪 (𝔽_{}^{})$ ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y = Y^{'} 1_{[0, τ)}^{} + Y^{''} (τ) 1_{[τ, \infty)}^{} .$ Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random times ...

Exact null internal controllability for the heat equation on unbounded convex domains

Viorel Barbu (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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The liner parabolic equation $\frac{y}{t} - \frac{1}{2} 𝔻 y + F \cdot y = {\vec{1}}_{_{0}} u$ ∂y ∂t − 1 2 Δy + F · ∇ y = 1 x1d4aa; 0 u with Neumann boundary condition on a convex open domain x1d4aa; ⊂ ℝ with smooth boundary is exactly null controllable on each finite interval if 𝒪is an open subset of x1d4aa; which contains a suitable neighbourhood of the recession cone of x1d4aa; . Here, : ℝ → ℝ is a bounded, -continuous function, and = ∇, where is convex and coercive.

Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

Hedy Attouch, Paul-Émile Maingé (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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In the setting of a real Hilbert space $ℋ$ , we investigate the asymptotic behavior, as time goes to infinity, of trajectories of second-order evolution equations () + $\dot{u}$ () + (()) + (()) = 0, where is the gradient operator of a convex differentiable potential function : $ℋ \to ℝ$ ,: $ℋ \to ℋ$ is a maximal monotone operator which is assumed to be-cocoercive, and > 0 is a damping parameter. Potential and non-potential effects are associated...

Some problems in automata theory which depend on the models of set theory

Olivier Finkel (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an -language $L (𝒜)$ (x1d49c;) accepted by a Büchi 1-counter automaton $𝒜$ x1d49c;. We prove the following surprising result: there exists a 1-counter Büchi automaton $𝒜$ x1d49c; such that the cardinality of the complement $L {(𝒜)}^{-}$ (𝒜) of the -language...

Higher-order phase transitions with line-tension effect

Bernardo Galvão-Sousa (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [ 4 (1987) 487–512], and in a different form by Alberti in [ is a scalar density function and and are double-well potentials, the exact scaling...

A new H(div)-conforming p-interpolation operator in two dimensions

Alexei Bespalov, Norbert Heuer (2011)

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

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In this paper we construct a new H(div)-conforming projection-based -interpolation operator that assumes only H() $\cap$ $\tilde{𝐇}$ (div, )-regularity ( > 0) on the reference element (either triangle or square) . We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space $\tilde{𝐇}$ (div, ), which is closely related...

Expansions for the distribution of M-estimates with applications to the Multi-Tone problem

Christopher S. Withers, Saralees Nadarajah (2011)

ESAIM: Probability and Statistics

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We give a stochastic expansion for estimates $\hat{θ}$ that minimise the arithmetic mean of (typically independent) random functions of a known parameter. Examples include least squares estimates, maximum likelihood estimates and more generally -estimates. This is used to obtain leading cumulant coefficients of $\hat{θ}$ needed for the Edgeworth expansions for the distribution and density ) to magnitude (or to for the symmetric...

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

Hang Yu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients , in three dimensions, $div (μ (\nabla u + \nabla u^{t})) + \nabla (λ div u) + V u = 0$ where and are Lipschitz continuous and . The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space (ℝ)

T. Suslina (2010)

Mathematical Modelling of Natural Phenomena

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In (ℝ; ℂ), we consider a wide class of matrix elliptic second order differential operators $𝒜$ with rapidly oscillating coefficients (depending on /). For a fixed > 0 and small > 0, we find approximation of the operator exponential exp(− $𝒜$ ) in the ( (ℝ; ℂ) → (ℝ; ℂ))-operator norm...

Exact simulation for solutions of one-dimensional Stochastic Differential Equations with discontinuous drift

Pierre Étoré, Miguel Martinez (2014)

ESAIM: Probability and Statistics

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In this note we propose an exact simulation algorithm for the solution of (1) $d X_{t} = d W_{t} + \bar{b} (X_{t}) d t, X_{0} = x,$ d X t = d W t + b̅ ( X t ) d t, X 0 = x, where $\bar{b}$ b̅is a smooth real function except at point 0 where $\bar{b} (0 +) \neq \bar{b} (0 -)$ b̅(0 + ) ≠ b̅(0 −) . The main idea is to sample an exact skeleton of Xusing an algorithm deduced from the convergence of the solutions of the skew...

Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations

Antoine Gloria (2012)

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

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We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the -dimensional lattice $ℤ^{d}$ with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error made by approximating the homogenized coefficient by the averaged energy of a regularized corrector...

Continuity of solutions of a nonlinear elliptic equation

Pierre Bousquet (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider a nonlinear elliptic equation of the form div [(∇)] + [] = 0 on a domain Ω, subject to a Dirichlet boundary condition tr = . We do not assume that the higher order term satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and satisfies a one-sided bounded slope condition, or when is radial: $a (ξ) = l (| ξ |) | ξ | ξ$ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasing:ℝ → ℝ

A bayesian framework for the ratio of two Poisson rates in the context of vaccine efficacy trials

Stéphane Laurent, Catherine Legrand (2012)

ESAIM: Probability and Statistics

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In many applications, we assume that two random observations and are generated according to independent Poisson distributions $(λ S)$ x1d4ab;() and $(μ T)$ x1d4ab;() and we are interested in performing statistical inference on the ratio = / of the two incidence rates. In vaccine efficacy trials, and are typically the numbers of cases in the vaccine and the control groups respectively, is called the relative risk and the statistical model is called ‘partial immunity model’. In this paper we...

Polynomial deviation bounds for recurrent Harris processes having general state space

Eva Löcherbach, Dasha Loukianova (2013)

ESAIM: Probability and Statistics

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Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using...

Surface energies in a two-dimensional mass-spring model for crystals

Florian Theil (2011)

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

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We study an atomistic pair potential-energy () that describes the elastic behavior of two-dimensional crystals with atoms where $y \in ℝ^{2 \times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of admits an asymptotic expansion involving fractional powers of : $\min_{y} E^{(n)} (y) = n E_{bulk} + \sqrt{n} E_{surface} + o (\sqrt{n}), n \to \infty .$ The bulk energy density ...

Coarse quantization for random interleaved sampling of bandlimited signals

Alexander M. Powell, Jared Tanner, Yang Wang, Özgür Yılmaz (2012)

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

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The compatibility of unsynchronized interleaved uniform sampling with Sigma-Delta analog-to-digital conversion is investigated. Let be a bandlimited signal that is sampled on a collection of interleaved grids { + } with offsets ${T_{n}}_{n = 1}^{N} \subset [0, T]$ T n n = 1 N ⊂ [ 0 ,T ] . If the offsets are chosen independently and uniformly at random from [0] and if the sample values of are quantized with a first order Sigma-Delta algorithm, then with high probability the quantization...

A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Andrew Lorent (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain ⊂ ℝthe functional is $I_{} (u) = \frac{1}{2} \int_{Ω}^{- 1} {| 1 - | D u |}^{2} |^{2} + {| D^{2} u |}^{2} d z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z wherebelongs to the subset of functions in $W_{0}^{2, 2} (Ω)$ W02,2(Ω) whose gradient (in the sense of trace) satisfies()· = 1 where is the inward pointing unit normal to at . In [1...