2
$\begingroup$

recently i am working in the following question:

Let $F(x,t)$ a Caratheodory function (i.e. $t \mapsto F(x,t)$ is continuous for all $x$ and $x \mapsto F(x,t)$ is Lebesgue mensurable for all $t$ ) such that

$$ |F(x,t)| \leq |t|^{2^\ast},\text{ for all } x\in \mathbb{R}^N, t\in \mathbb{R}, $$ for some $C>0.$

Is it true that the functional

$$ G(u) = \int _{\mathbb{R}^N} F(x,u(x)) dx, u \in D^{1,2} (\mathbb{R}^N), $$

is uniformly continuous in bounded sets of $D^{1,2} (\mathbb{R}^N)$?

If not, what is the assumptions made over $F$ that guarantees the affirmative for the question?

For instance, if $F_t$ exists (the partial derivative in $t$ exists ) and it is Caratheodory, with

$$ |F_t (x,t)| \leq |t|^{2^\ast-1},\text{ for all } x\in \mathbb{R}^N, t\in \mathbb{R}, $$ then, by the Holder's inequality and the Mean Value Theorem, the answer for the question is yes.

In any case, i would appreciate any literature indications.

Thank you for the attention.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.