# From some priori estimates can we estimate higher Sobolev norm?

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.

• 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. Aug 17, 2021 at 20:28
• 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. Aug 17, 2021 at 20:29
• 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?
– Keba
Aug 17, 2021 at 21:58