Asymptotic stability of Schrödinger semigroups: path integral methods.
In the recent years, many results have been established on positive solutions for boundary value problems of the form 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).
We prove that minimizers of the functional , ⊂ , n ≥ 3, which satisfy the Dirichlet boundary condition on for g: → with zero topological degree, converge in and for any α<1 - upon passing to a subsequence - 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.
Let (x,u,∇u) be a Lagrangian periodic of period 1 in x1,...,xn,u. We shall study the non self intersecting functions u: RnR 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) + jx. Moser has shown that each of these functions is at finite distance from a plane u = ρx 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 since...