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Let $\Omega\subset \mathbb{R}^n, N\geq3$, be an open set. For $p\in(1,+\infty)$, define a functional $J:W_0^{1,p}(\Omega)\rightarrow\mathbb{R}$ by

$J(u)=\int_\Omega |\nabla u|^p\,dx.$

Then $J$ is differentiable in $W_0^{1,p}(\Omega)$ and

$J'(u)v=p\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla v\,dx.$

My question is about how to prove the Gateaux derivative $J_G'(u)$ is continuous. Indeed, for any sequence $u_k\rightarrow u$, I can prove that there exists a subsequence $u_{k_j}$ satisfying $J_G'(u_{k_j})\rightarrow J_G'(u)$ in [$W^{1,p}(\Omega)$]', but I can't prove it for the original sequence.

Thanks for any help!

Kimura Leo
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1 Answers1

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Use the following convergence principle:

A sequence is converging to some $x$ if and only if every subsequence contains a subsequence that converges to $x$.

daw
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  • This is a very useful principle, well worth memorizing. – MaoWao Mar 23 '21 at 10:25
  • Could you tell me how to prove this principle? I can not fully understand it. Thanks! – Kimura Leo Mar 27 '21 at 06:36
  • By contradiction: assume the sequence does not converge to $x$, then there is a subsequence such that all elements have distance larger than $\epsilon>0$ to $x$, ... – daw Mar 27 '21 at 10:12