Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions

Paweł Strzelecki

Displaying similar documents to “Asymptotics for the minimization of a Ginzburg-Landau energy in n dimensions”

On $\infty$ -harmonic functions

Thierry De Pauw (2007)

Journées Équations aux dérivées partielles

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On the solvability of nonlinear elliptic equations in Sobolev spaces

Piotr Fijałkowski (1992)

Annales Polonici Mathematici

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We consider the existence of solutions of the system (*) $P (D) u^{l} = F (x, (\partial^{α} u))$ , l = 1,...,k, $x \in ℝ^{n}$ $(u = (u ¹, . . ., u^{k}))$ in Sobolev spaces, where P is a positive elliptic polynomial and F is nonlinear.

A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization

Chunmei Wang (2014)

Applications of Mathematics

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In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by ${(1 + log (H / h))}^{2}$ , where $H$ and $h$ are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.

On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions

Joachim Naumann, Jörg Wolf, Michael Wolff (1998)

Commentationes Mathematicae Universitatis Carolinae

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We prove the interior Hölder continuity of weak solutions to parabolic systems $\frac{\partial u^{j}}{\partial t} - D_{α} a_{j}^{α} (x, t, u, \nabla u) = 0 in Q (j = 1, ..., N)$ ( $Q = Ω \times (0, T), Ω \subset ℝ^{2}$ ), where the coefficients $a_{j}^{α} (x, t, u, ξ)$ are measurable in $x$ , Hölder continuous in $t$ and Lipschitz continuous in $u$ and $ξ$ .

Existence of solutions of degenerated unilateral problems with $L^{1}$ data

Lahsen Aharouch, Youssef Akdim (2004)

Annales mathématiques Blaise Pascal

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In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type $A u + g (x, u, \nabla u) = f - div F,$ where $A$ is a Leray-Lions operator and $g$ is a Carathéodory function having natural growth with respect to $| \nabla u |$ and satisfying the sign condition. The second term is such that, $f \in L^{1} (Ω)$ and $F \in Π_{i = 1}^{N} L^{p^{'}} (Ω, w_{i}^{1 - p^{'}})$ .

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

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1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x = 1 / d ₁ (x) + 1 / (d ₁ (x) d ₂ (x)) + . . . + 1 / (d ₁ (x) d ₂ (x) . . . d_{n} (x)) + . . .$ , where $d_{j} (x), j \geq 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j + 1} (x) \geq d_{j} (x)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $l i m_{n \to \infty} d_{n}^{1 / n} (x) = e .$ He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $d i m_{H} x \in (0, 1] : (2) f a i l s = 1$ . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $d i m_{H}$ to denote...

Inessentiality with respect to subspaces

Michael Levin (1995)

Fundamenta Mathematicae

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Let X be a compactum and let $A = (A_{i}, B_{i}) : i = 1, 2, . . .$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_{i}$ separating $A_{i}$ and $B_{i}$ the intersection $(\cap F_{i}) \cap Y$ is not empty. So A is inessential on Y if there exist closed $F_{i}$ separating $A_{i}$ and $B_{i}$ such that $\cap F_{i}$ does not intersect Y. Properties of inessentiality are studied and applied to prove: Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

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Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map $g : X \to 𝕀^{k}$ such that dim (f × g) = 1. We improve this result of Sternfeld...