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

$W:\mathrm{\Omega }\times {\mathbb{ℝ}}^N\to \mathbb{ℝ}\cup \{+\mathrm{\infty }\}$, for $\mathrm{\Omega }\subseteq {\mathbb{ℝ}}^d$ endowed with the Lebesgue measure, is a Carathéodory function if:

1. The mapping $x\mapsto W(x,\xi )$ is Lebesgue-measurable for every $\xi \in {\mathbb{ℝ}}^N$.

2. the mapping $\xi \mapsto W(x,\xi )$ is continuous for almost every $x\in \mathrm{\Omega }$.

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

Example

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

p-growth

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