Bounded sequence in Sobolev Space

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'.

• What is $\Delta$? – timur Dec 26 '18 at 19:43
• @timur Sorry I meant to write $||Du||_{W^{1,4}(\mathbb{T}^2)}$, I will edit now – 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 $$\alpha,\beta$$.
• Is it obvious that we can apply Poincare-Wirtinger in this way for all $u \in W^{1,4}(\mathbb{T}^2)$ – rogerroger Dec 26 '18 at 21:05
• @rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}\subset W^{1,2}$. – timur Dec 26 '18 at 21:10
• and I assume you have used $u$ having zero average by not regarding it in your $||\cdot ||_{W^{1,4}}$ norm? – rogerroger Dec 27 '18 at 17:43