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Hi I have the next claim which I would like to find a proof of it.

I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,H^1(M))\cap Lip(I,L^2(M))$ where $I$ is some interval that includes zero in it.

The claim is that $\partial_t u_\epsilon$ is bounded in $L^\infty(I,L^2(M))$, why is that necessarily true?

Thanks in advance.

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  • $\begingroup$ Because that is the same as Lip. $\endgroup$
    – Michael Renardy
    Apr 18, 2015 at 15:46
  • $\begingroup$ Can you elaborate, I am not sure I follow. BTW, I edited and changed $L^2(M)$ to $H^1(M)$ in the space $L^\infty(I,L^2(M))\cap Lip(I,L^2(M))$. $\endgroup$
    – Alan
    Apr 18, 2015 at 15:54

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