Sobolev Interpolation problem

Question

I was asked to prove $$\|u\|_{W^{1,4}(\Omega)}\le C\|u\|_{W^{2,2}(\Omega)}^\theta\|u\|_{L^2(\Omega)}^{1-\theta}$$

here $\Omega\subset\mathbb R^2$

but I don't know where to start, is there something to do with the sobolev inequality?

I only know how to calculate the index: by scaling argument $\theta=3/4$.

@gerw In Adam's "Sobolev Spaces" $|u|{W^{j,p}}\le C|u|{W^{m,p}}^\theta|u|_{L^p}^{1-\theta}$ when $j\le m$, but the index doesn't match here. — Lookout, Nov 20 '16 at 01:18

score 1 · Accepted Answer · answered Nov 20 '16 at 18:30

First use the embedding of $H^1$ into some $L^p$, $p>4$. Then interpolate $L^p$-norms (H\"older inequality), then apply the result you mentioned: $$ \|v\|_{L^p} \le c \|v\|_{H^1}, $$ $$ \|v\|_{L^4} \le \|v\|_{L^2}^\theta \|v\|_{L^p}^{1-\theta} \le c \|v\|_{L^2}^\theta \|v\|_{H^1}^{1-\theta} $$ Since you are dealing with 2-dimensional domain, this is true with $\theta=1/2$ (see Temam's book on Navier-Stokes). Set $v:=\nabla u$. Then we get $$ \|\nabla u\|_{L^4} \le c \|u\|_{H^1}^\theta \|u\|_{H^2}^{1-\theta} \le c \|u\|_{L^2}^{\theta'} \|u\|_{H^2}^{1-\theta'} $$ where in the last step I used the interpolation inequality for Sobolev spaces you mentioned in the comment. As written in the question, $1-\theta'=3/4$.

Sobolev Interpolation problem

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