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

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

1. The mapping   is Lebesgue-measurable for every  .

2. the mapping   is continuous for almost every  .

The main merit of Carathéodory function is the following: If   is a Carathéodory function and   is Lebesgue-measurable, then the composition   is Lebesgue-measurable.[1]

Example

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Many problems in the calculus of variation are formulated in the following way: find the minimizer of the functional   where   is the Sobolev space, the space consisting of all function   that are weakly differentiable and that the function itself and all its first order derivative are in  ; and where   for some  , a Carathéodory function. The fact that   is a Carathéodory function ensures us that   is well-defined.

p-growth

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If   is Carathéodory and satisfies   for some   (this condition is called "p-growth"), then   where   is finite, and continuous in the strong topology (i.e. in the norm) of  .

References

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