MathDB

Let

U\subset\mathbb R^n

be an open set, and let

L: U\times\mathbb R^n\to\mathbb R

be a continuous, in its second variable first order positive homogeneous, positive over

U\times (\mathbb R^n\setminus\{0\})

and of

C^2

-class Langrange function, such that for all

p\in U

the Gauss-curvature of the hyper surface \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \} is nowhere zero. Determine the extremals of

L

if it satisfies the following system \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\}) of partial differetial equations, where y^k(u,v) : \equal{} v^k for

(u,v)\in U\times\mathbb R^k

, v \equal{} (v^1,\dots,v^k).

Problem Statement