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Asymptotics for quasilinear elliptic non-positone problems

Zuodong Yang, Qishao Lu (2002)

Annales Polonici Mathematici

In the recent years, many results have been established on positive solutions for boundary value problems of the form - d i v ( | u ( x ) | p - 2 u ( x ) ) = λ f ( u ( x ) ) in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).

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 , ⊂ 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 (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 ; 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...

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