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Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that $$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$ where $u|_{\partial\Omega}=\phi$.

Can we estimate $L^r$-norm of $u$ for higher $r$ in terms of some norm in $\phi$ ? More specifically I am interested to know if $\|u\|_{L^{2(p-1)}}$ is bounded by some $L^r(\partial\Omega)$-norm of $\phi$?

Any help or reference will be greatly appreciated.

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    $\begingroup$ As stated: I don't think so. It shouldn't be too hard to come up with counterexamples. If $2(p-1)$ is outside the Sobolev embedding range you can add compactly supported perturbations to $u$ that doesn't change its $W^{1,p}$ norm much but makes the $L^{2(p-1)}$ norm arbitrary large. $\endgroup$ Aug 17, 2021 at 20:28
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    $\begingroup$ Alternatively, think about $\phi$ which is highly oscillatory but has small amplitude. // You need a more definite connection between $\phi$ and $u$ for something like what you want to potentially work. $\endgroup$ Aug 17, 2021 at 20:29
  • $\begingroup$ Does $u$ solve some equation? If not, is not every nontrivial function with compact support in $\Omega$ a counterexample to both statements? Or should the inequality be the other way around? $\endgroup$
    – Keba
    Aug 17, 2021 at 21:58

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