Английская Википедия:Carathéodory function

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In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory.

Definition

<math> W:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R}\cup\left\{ +\infty\right\} </math>, for <math> \Omega\subseteq\mathbb{R}^{d} </math> endowed with the Lebesgue measure, is a Carathéodory function if:

1. The mapping <math> x\mapsto W\left(x,\xi\right) </math> is Lebesgue-measurable for every <math> \xi\in\mathbb{R}^{N} </math>.

2. the mapping <math> \xi\mapsto W\left(x,\xi\right) </math> is continuous for almost every <math> x\in\Omega </math>.

The main merit of Carathéodory function is the following: If <math> W:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R} </math> is a Carathéodory function and <math> u:\Omega\rightarrow\mathbb{R}^{N} </math> is Lebesgue-measurable, then the composition <math> x\mapsto W\left(x,u\left(x\right)\right) </math> is Lebesgue-measurable.

Example

Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional <math> \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R}\cup\left\{ +\infty\right\} </math> where <math> W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) </math> is the Sobolev space, the space consisting of all function <math> u:\Omega\rightarrow\mathbb{R}^{m} </math> that are weakly differentiable and that the function itself and all its first order derivative are in <math> L^{p}\left(\Omega;\mathbb{R}^{m}\right) </math>; and where <math> \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx </math> for some <math> W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} </math>, a Carathéodory function. The fact that <math> W </math> is a Carathéodory function ensures us that <math> \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx </math> is well-defined.

p-growth

If <math> W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} </math> is Carathéodory and satisfies <math> \left|W\left(x,v,A\right)\right|\leq C\left(1+\left|v\right|^{p}+\left|A\right|^{p}\right) </math> for some <math> C>0 </math> (this condition is called "p-growth"), then <math> \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R} </math> where <math> \mathcal{F}\left[u\right]=\int_{\Omega}W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx </math> is finite, and continuous in the strong topology (i.e. in the norm) of <math> W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) </math>.