Critical points via $\Gamma $-convergence: general theory and applications

Jerrard, Robert L.; Peter Sternberg

Displaying similar documents to “Critical points via $Γ$ -convergence: general theory and applications”

Bulk superconductivity in Type II superconductors near the second critical ﬁeld

Soren Fournais, Bernard Helffer (2010)

Journal of the European Mathematical Society

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We consider superconductors of Type II near the transition from the ‘bulk superconducting’ to the ‘surface superconducting’ state. We prove a new $L^{\infty}$ estimate on the order parameter in the bulk, i.e. away from the boundary. This solves an open problem posed by Aftalion and Serfaty [AS].

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $- Δ u + u^{3} + β u v^{2} = λ u, - Δ v + v^{3} + β u^{2} v = μ v, u, v \in H_{0}^{1} (Ω), u, v > 0$ , as the interspecies scattering length $β$ goes to $+ \infty$ . For this system we consider the associated energy functionals $J_{β}, β \in (0, + \infty)$ , with $L^{2}$ -mass constraints, which limit $J_{\infty}$ (as $β \to + \infty$ ) is strongly irregular. For such functionals, we construct multiple critical points...

The Cauchy problem for the liquid crystals system in the critical Besov space with negative index

Sen Ming, Han Yang, Zili Chen, Ls Yong (2017)

Czechoslovak Mathematical Journal

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The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p, 1}^{n / p - 1} (ℝ^{n}) \times {\dot{B}}_{p, 1}^{n / p} (ℝ^{n})$ with $n < p < 2 n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

A strong convergence theorem for H¹(𝕋ⁿ)

Feng Dai (2006)

Studia Mathematica

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Let ⁿ denote the usual n-torus and let $S ̃_{u}^{δ} (f)$ , u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $l i m_{R \to \infty} 1 / l o g R \int_{0}^{R} (| | S ̃_{u}^{α} (f) - {f | |}_{H ¹ (ⁿ)}) / (u + 1) d u = 0$ .

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

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In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ $- (a + b \int_{Ω} | \nabla u | ² d x) Δ u = u ⁵ + λ u^{q - 1} / {| x |}^{β}$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

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We study the following singular elliptic equation with critical exponent ⎧ $- Δ u = Q (x) u^{2 * - 1} + λ u^{- γ}$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

On cusps and flat tops

Neil Dobbs (2014)

Annales de l’institut Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1 + ϵ}$ . The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1 + ϵ}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

Linearized plasticity is the evolutionary $Γ$ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $Γ$ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

The Energy Density of Non Simple Materials Grade Two Thin Films via a Young Measure Approach

Giuliano Gargiulo, Elvira Zappale (2007)

Bollettino dell'Unione Matematica Italiana

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Dimension reduction is used to derive the energy of non simple materials grade two thin films. Relaxation and $\Gamma$ convergence lead to a limit defined on a suitable space of bi-dimensional Young measures. The underlying ``deformation'' in the limit model takes into account the Cosserat theory.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

J. Chabrowski, Shusen Yan (2002)

Colloquium Mathematicae

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We consider the Neumann problem for the equation $- Δ u - λ u = {Q (x) | u |}^{2 * - 2} u$ , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k - 1}$ and $λ_{k}$ . Applying a min-max principle based on topological linking we prove the existence of a solution.

On asymptotic critical values and the Rabier Theorem

Zbigniew Jelonek (2004)

Banach Center Publications

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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f ₁, . . ., f_{m}) : X \to k^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with...

Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Y. Akdim, J. Bennouna, M. Mekkour, H. Redwane (2012)

Applicationes Mathematicae

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We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), ${b (x, u) |}_{t = 0} = b (x, u ₀)$ in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in $L ¹ (Q) + L^{p^{'}} (0, T; W^{- 1, p^{'}} (Ω))$ and b(x,u₀) ∈ L¹(Ω).

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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On a class of nonlocal problem involving a critical exponent

Anass Ourraoui (2015)

Communications in Mathematics

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In this work, by using the Mountain Pass Theorem, we give a result on the existence of solutions concerning a class of nonlocal $p$ -Laplacian Dirichlet problems with a critical nonlinearity and small perturbation.

Lieb–Thirring inequalities on the half-line with critical exponent

Tomas Ekholm, Rupert Frank (2008)

Journal of the European Mathematical Society

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We consider the operator $- d^{2} / d r^{2} - V$ in $L_{2} (ℝ_{+})$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $tr {(- d^{2} / d r^{2} - V)}_{-}^{γ} \leq C_{γ, α} \int_{ℝ_{+}} {(V (r) - 1 / (4 r^{2}))}_{+}^{γ + (1 + α) / 2} r^{α} d r$ for any $α \in [0, 1)$ and $γ \geq (1 - α) / 2$ . This includes a Lieb-Thirring inequality in the critical endpoint case.

Estimating the critical determinants of a class of three-dimensional star bodies

Werner Georg Nowak (2017)

Communications in Mathematics

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In the problem of (simultaneous) Diophantine approximation in $ℝ^{3}$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $K_{2} : (y^{2} + z^{2}) (x^{2} + y^{2} + z^{2}) \leq 1$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $Δ (K_{c})$ of more general star bodies $K_{c} : {(y^{2} + z^{2})}^{c / 2} (x^{2} + y^{2} + z^{2}) \leq 1,$ where $c$ is any positive constant. These are obtained by inscribing into $K_{c}$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of...

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

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Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ ${Δ (| Δ u |}^{p - 2} {Δ u) - d i v (| \nabla u |}^{p - 2} \nabla u) = λ f (x, u) + μ g (x, u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

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The role of the second critical exponent $p = (n + 1) / (n - 3)$ , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $Δ u + u^{p} = 0$ , $u > 0$ under zero Dirichlet boundary conditions, in a domain $Ω$ in $ℝ^{n}$ with bounded, smooth boundary. Given $Γ$ , a geodesic of the boundary with negative inner normal curvature we find that for $p = (n + 1) / (n - 3 - ε)$ , there exists a solution $u_{ε}$ such that $| \nabla u_{ε} |^{2}$ converges weakly to a Dirac measure on $Γ$ as $ε \to 0^{+}$ , provided that $Γ$ is nondegenerate in the sense of second...

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

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We study the leading order behaviour of positive solutions of the equation $- Δ u + ϵ u - {| u |}^{p - 2} u + {| u |}^{q - 2} u = 0, x \in ℝ^{N}$ , where $N \geq 3$ , $q > p > 2$ and when $ϵ > 0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$ , $q$ and $N$ . The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $2^{*} = \frac{2 N}{N - 2}$ . For $p < 2^{*}$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p > 2^{*}$ the solution asymptotically...

Displaying similar documents to “Critical points via Γ -convergence: general theory and applications”

Displaying similar documents to “Critical points via $Γ$ -convergence: general theory and applications”