Carathéodory function explained

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 x R^N → R\cup\left\{+infty\right\}

, for

\Omega\subseteqR^d

endowed with the Lebesgue measure, is a Carathéodory function if:

1. The mapping

x\mapstoW\left(x,\xi\right)

is Lebesgue-measurable for every

\xi\inR^N

2. the mapping

\xi\mapstoW\left(x,\xi\right)

is continuous for almost every

x\in\Omega

The main merit of Carathéodory function is the following: If

W:\Omega x R^N → R

is a Carathéodory function and

u:\Omega → R^N

is Lebesgue-measurable, then the composition

x\mapstoW\left(x,u\left(x\right)\right)

is Lebesgue-measurable.^[1]

Example

Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional

l{F}:W^1,p\left(\Omega;R^m\right) → R\cup\left\{+infty\right\}

where

W^1,p\left(\Omega;R^m\right)

is the Sobolev space, the space consisting of all function

u:\Omega → R^m

that are weakly differentiable and that the function itself and all its first order derivative are in

L^p\left(\Omega;R^m\right)

; and where

l{F}\left[u\right]=\int_\OmegaW\left(x,u\left(x\right),\nablau\left(x\right)\right)dx

for some

W:\Omega x R^m x R^{d x} → R

, a Carathéodory function.The fact that

is a Carathéodory function ensures us that

l{F}\left[u\right]=\int_\OmegaW\left(x,u\left(x\right),\nablau\left(x\right)\right)dx

is well-defined.

p-growth

W:\Omega x R^m x R^{d x} → R

is Carathéodory and satisfies

for some

C>0

(this condition is called "p-growth"), then

l{F}:W^1,p\left(\Omega;R^m\right) → R

where

l{F}\left[u\right]=\int_\OmegaW\left(x,u\left(x\right),\nablau\left(x\right)\right)dx

is finite, and continuous in the strong topology (i.e. in the norm) of

W^1,p\left(\Omega;R^m\right)

Notes and References

Rindler, Filip (2018). Calculus of Variation. Springer Cham. p. 26-27.