If $u_k\rightharpoonup u$ weakly in $W^{1,q}(U)$, how can I show $$ \sup\limits_{k}||u_k||_{W^{1.q}(U)}<\infty? $$
2026-04-12 22:48:11.1776034091
How to show $ \sup\limits_{k}||u_k||_{W^{1.q}(U)}<\infty $?
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I second @Jose27's answer. This is the general fact from functional analysis. For a proof, please see Proposition 3.5 in this book.
Moreover, the converse is also true as well. If a sequence of function is uniformly bounded in $W^{1,p}(\Omega)$ for $\Omega$ bounded and satisfies some boundary regularity, then, up to a subsequence, you can extract a weak convergence sequence.