Carathéodory function

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

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

2. the mapping [math]\displaystyle{ \xi\mapsto W\left(x,\xi\right) }[/math] is continuous for almost every [math]\displaystyle{ x\in\Omega }[/math].

The main merit of Carathéodory function is the following: If [math]\displaystyle{ W:\Omega\times\mathbb{R}^{N}\rightarrow\mathbb{R} }[/math] is a Carathéodory function and [math]\displaystyle{ u:\Omega\rightarrow\mathbb{R}^{N} }[/math] is Lebesgue-measurable, then the composition [math]\displaystyle{ 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]\displaystyle{ \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R}\cup\left\{ +\infty\right\} }[/math] where [math]\displaystyle{ W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) }[/math] is the Sobolev space, the space consisting of all function [math]\displaystyle{ 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]\displaystyle{ L^{p}\left(\Omega;\mathbb{R}^{m}\right) }[/math]; and where [math]\displaystyle{ \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]\displaystyle{ W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} }[/math], a Carathéodory function. The fact that [math]\displaystyle{ W }[/math] is a Carathéodory function ensures us that [math]\displaystyle{ \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]\displaystyle{ W:\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{d\times m}\rightarrow\mathbb{R} }[/math] is Carathéodory and satisfies [math]\displaystyle{ \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]\displaystyle{ C\gt 0 }[/math] (this condition is called "p-growth"), then [math]\displaystyle{ \mathcal{F}:W^{1,p}\left(\Omega;\mathbb{R}^{m}\right)\rightarrow\mathbb{R} }[/math] where [math]\displaystyle{ \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]\displaystyle{ W^{1,p}\left(\Omega;\mathbb{R}^{m}\right) }[/math].