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.

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

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

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

The main merit of Carathéodory function is the following: If ${\displaystyle W:\Omega \times \mathbb {R} ^{N}\rightarrow \mathbb {R} }$ is a Carathéodory function and ${\displaystyle u:\Omega \rightarrow \mathbb {R} ^{N}}$ is Lebesgue-measurable, then the composition ${\displaystyle x\mapsto W\left(x,u\left(x\right)\right)}$ is Lebesgue-measurable.^[1]

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

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