Strikte Konvexität für Variationsprobleme auf demn-dimensionalen Torus

Abstract

We consider a variational problem with an integrandF:R ⁿ×R×R ⁿ→R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R ⁿ minimal solutionsu:R ⁿ→R minimising ƒF(x, u(x), u _x (x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action$M(a) = \lim _{r \to \infty } \frac{1}{{volB_r }}\int_{B_r } {F(x,u,u_x )dx}$ (whereB _r is ther-ball around 0∈R ⁿ) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.