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

Paweł Strzelecki (1996)

Colloquium Mathematicae

We prove that minimizers u ∈ W 1 , n of the functional E ( u ) = 1 / n ∫ | ∇ u | n d x + 1 / ( 4 n ) ∫ ( 1 - | u | 2 ) 2 d x , ⊂ ℝ n , n ≥ 3, which satisfy the Dirichlet boundary condition u = g on for g: → S n - 1 with zero topological degree, converge in W 1 , n and C l o c α for any α<1 - upon passing to a subsequence k → 0 - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.

Aubry sets and the differentiability of the minimal average action in codimension one

Ugo Bessi (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Let ℒ (x,u,∇u) be a Lagrangian periodic of period 1 in x1,...,xn,u. We shall study the non self intersecting functions u: Rn → R minimizing ℒ ; non self intersecting means that, if u(x0 + k) + j = u(x0) for some x0∈Rn and (k , j) ∈Zn × Z, then u(x) = u(x + k) + j ∀ x. Moser has shown that each of these functions is at finite distance from a plane u = ρ · x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called β ( ρ ) since...

Bifurcations for a problem with jumping nonlinearities

Lucie Kárná, Milan Kučera (2002)

Mathematica Bohemica

A bifurcation problem for the equation Δ u + λ u - α u + + β u - + g ( λ , u ) = 0 in a bounded domain in N with mixed boundary conditions, given nonnegative functions α , β ∈ L ∞ and a small perturbation g is considered. The existence of a global bifurcation between two given simple eigenvalues λ ( 1 ) , λ ( 2 ) of the Laplacian is proved under some assumptions about the supports of the functions α , β . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to λ ( 1 ) , λ ( 2 ) .

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