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:\Omega\times\mathbb^\rightarrow\mathbb\cup\left\$, for $\Omega\subseteq\mathbb^$ endowed with the Lebesgue measure, is a Carathéodory function if:

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

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

The main merit of Carathéodory function is the following: If $W:\Omega\times\mathbb^\rightarrow\mathbb$ is a Carathéodory function and $u:\Omega\rightarrow\mathbb^$ is Lebesgue-measurable, then the composition $x\mapsto W\left(x,u\left(x\right)\right)$ is Lebesgue-measurable.Rindler, Filip (2018). ''Calculus of Variation''. Springer Cham. p. 26-27.

Example

Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional $\mathcal:W^\left(\Omega;\mathbb^\right)\rightarrow\mathbb\cup\left\$ where $W^\left(\Omega;\mathbb^\right)$ is the Sobolev space, the space consisting of all function $u:\Omega\rightarrow\mathbb^$ that are weakly differentiable and that the function itself and all its first order derivative are in $L^\left(\Omega;\mathbb^\right)$; and where \mathcal\left \right\int_W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx for some $W:\Omega\times\mathbb^\times\mathbb^\rightarrow\mathbb$, a Carathéodory function.
The fact that $W$ is a Carathéodory function ensures us that \mathcal\left \right\int_W\left(x,u\left(x\right),\nabla u\left(x\right)\right)dx is well-defined.

p-growth

If $W:\Omega\times\mathbb^\times\mathbb^\rightarrow\mathbb$ is Carathéodory and satisfies $\left, W\left(x,v,A\right)\\leq C\left(1+\left, v\^+\left, A\^\right)$ for some $C>0$ (this condition is called "p-growth"), then $\mathcal:W^\left(\Omega;\mathbb^\right)\rightarrow\mathbb$ where \mathcal\left \right\int_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 $W^\left(\Omega;\mathbb^\right)$.

References

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Calculus of variations