1
$\begingroup$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.

If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like estimate on the $W^{1,2}$-norm of $f$) is sufficient to obtain that $f \in L^\infty(I \times \Omega)$?

$\endgroup$
  • $\begingroup$ Can you give some idea of what sort of $L^p$ estimates you have in mind? $\endgroup$ – Nate Eldredge Dec 11 '18 at 23:36
  • $\begingroup$ @NateEldredge Honestly I don't have a precise idea. $\endgroup$ – Rene Dec 12 '18 at 1:19
  • $\begingroup$ Well, doesn't $W^{1,2}(U)$ contains functions like $\ln(x^2+y^2)$, which should be in every $L^p$ but unbounded? So I don't see how any simple sort of $L^p$ bound is going to help at all. $\endgroup$ – Nate Eldredge Dec 12 '18 at 1:34
  • $\begingroup$ @NateEldredge That's true. Maybe some decay-like estimate for some $L^p$ norms or the $H^1$ norm could help? $\endgroup$ – Rene Dec 12 '18 at 1:43
  • 1
    $\begingroup$ What is a decay-like condition on a norm? $\endgroup$ – Hannes Dec 12 '18 at 10:37

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.