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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...

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

Josef Dalík, Václav Valenta (2013)

Open Mathematics

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite...

Averaging techniques and oscillation of quasilinear elliptic equations

Zhi-Ting Xu, Bao-Guo Jia, Shao-Yuan Xu (2004)

Annales Polonici Mathematici

By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order i , j = 1 N D i [ A i j ( x ) | D y | p - 2 D j y ] + p ( x ) f ( y ) = 0 are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

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