Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :

Define : The Sobolev Space $W^{1,4}(\mathbb{T}^{2})$ as $$W^{1,4}(\mathbb{T}^{2})=\{u~:~~ u\in L^4_{loc}(\mathbb{R}^2), ~~ \Delta u\in L^4_{loc}(\mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ \int_{(0,1)^{2}}u~=0 \} $$

Define : The functional $J:W^{1,4}(\mathbb{T}^2)\to \mathbb{R} $ (for some $f\in L^2(\mathbb{T}^2) $)

$$J(u)=\frac{1}{2}\int_{(0,1)^{2}}|Du|^2 + \int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - \int_{(0,1)^{2}}fu$$

Show that if $J(u_{n})\to \inf_{u}J(u)$ then $\{ u_{n}\}$ is bounded in $W^{1,4}(\mathbb{T}^2)$.

$\bf{Hint}$ Use Poincare-Writinger inequality to justify why $\exists C>0$ s.t for any $u\in W^{1,4}(\mathbb{T}^2)$

$$||u||_{W^{1,4}(\mathbb{T}^2)}\leq C|| D u||_{W^{1,4}(\mathbb{T}^2)} $$

Note : $Du=(u_{x_{1}},u_{x_{2}})$

$\bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)

Let $\epsilon>0$ using the definition of limit $\exists N$ s.t $\forall n>N$

$$| J(u_{n})-\inf J(u) | \leq \epsilon $$

Now using definition of infimum it follows

$$J(u_{n})\leq \epsilon +\inf J(u) $$ $$J(u_{n}) \leq \eta ~~~~ \forall n>N $$

Now were done if we can show $||u_{n}||_{W^{1,4}(\mathbb{T}^2)} \leq J(u_{n})$ Im struggling to extract $||u_{n}||_{W^{1,4}}(\mathbb{T}^2)$ from the above because of the $-\int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.

  • $\begingroup$ What is $\Delta$? $\endgroup$ – timur Dec 26 '18 at 19:43
  • $\begingroup$ @timur Sorry I meant to write $||Du||_{W^{1,4}(\mathbb{T}^2)}$, I will edit now $\endgroup$ – rogerroger Dec 26 '18 at 19:59

We have $$ |\int fu| \leq \|f\|_{L^2}\|u\|_{L^2} \leq \varepsilon\|u\|_{L^2}^2+\frac4\varepsilon\|f\|_{L^2}^2 \leq C\varepsilon\|Du\|_{L^2}^2+\frac4\varepsilon\|f\|_{L^2}^2 , $$ where we have used the fact that $\int u=0$ in combination with the Poincare-Wirtinger inequality. Take $\varepsilon>0$ small enough, and get $$ \|Du\|_{L^4}^4\leq\alpha J(u) + \beta $$ for some constants $\alpha,\beta$.

  • $\begingroup$ Is it obvious that we can apply Poincare-Wirtinger in this way for all $u \in W^{1,4}(\mathbb{T}^2)$ $\endgroup$ – rogerroger Dec 26 '18 at 21:05
  • $\begingroup$ Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :) $\endgroup$ – rogerroger Dec 26 '18 at 21:06
  • 1
    $\begingroup$ @rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}\subset W^{1,2}$. $\endgroup$ – timur Dec 26 '18 at 21:10
  • $\begingroup$ and I assume you have used $u$ having zero average by not regarding it in your $||\cdot ||_{W^{1,4}}$ norm? $\endgroup$ – rogerroger Dec 27 '18 at 17:43
  • $\begingroup$ I don't know exactly which step you are referring to but you will need that at some point. $\endgroup$ – timur Dec 27 '18 at 19:50

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