Bounding the integral of a function by the integral of its derivative

Question

I have no idea where to begin for this question, so any help would be greatly appreciated!

Let $\Omega$ be a square with side 1. Show that $$\left(\int_{\Omega} v^2 \, dx \right)^{1/2} \leq \left( \int_{\Omega} |\nabla v|^2 \, dx \right)^{1/2}$$ for all $v \in H^1_0(\Omega)$.

The 1D analogue of this problem is really straightforward with the Fundamental Theorem of Calculus. I don't think that can be used for this problem again, so I'm a stuck.

Answer 1

Poincaré inequality holds for every subspace of $W^{1,p}(\Omega)$ which has compact embedding in $L^p$ and does not contain constants.

Let me be more precise.

Theorem. Let $\Omega$ be an open, Lipschitz, bounded, connected set in $\mathbb R^d$ and let $p \in [1,+\infty)$. Let $W \subset W^{1,p}$ be a subspace which has compact embedding in $L^p(\Omega)$ and does not contain constants. Then there exists a constant $C=C(\Omega,p)>0$ such that $$ \Vert u \Vert_{L^p(\Omega)} \le C \Vert \nabla u \Vert_{L^p(\Omega)} \qquad \forall u \in W. $$

Proof. By contradiction, suppose for every $n \in \mathbb N$ there exists a function $u_n \in W$ such that $$ \Vert u_n \Vert_{L^p(\Omega)} > n \Vert \nabla u_n \Vert_{L^p(\Omega)}. $$ Clearly, it is not restrictive to assume $\Vert u_n \Vert_{L^p(\Omega)} = 1$ for every $n \in \mathbb N$, hence we have $$ \Vert \nabla u_n \Vert_{L^p(\Omega)}< \frac{1}{n}. $$ In other words, the sequence $(u_n)_{n\in \mathbb N}$ is bounded in $W$; since by assumption the injection $W \hookrightarrow L^p$ is compact, we deduce that - up to subsequences - $u_n \to u$ in $L^p$. Furthermore, we also have $\nabla u_n \to 0$ and it is easy to see that $u \in W$, in particular $\nabla u = 0$. Thus $u$ is a constant function in $W$, which is a contradiction. QED

In particular, there are two nice subspaces satisfying hypothesis: $W_0^{1,p}$ (which gives what you are looking for in the case $p=2$) and $$ W_{\star}^{1,p} := \left\{u \in W^{1,p}: \, \int u = 0 \right\}, $$ i.e. the space of functions with zero average (this choice of $W$ gives the inequality you were speaking about in the comment).

Further comments.

1. I think there exist tons of proofs of Poincaré inequality; I suggest you to give a look to some classical book about Sobolev spaces, for instance Brezis' one (see Chap. 9, Cor. 9.19) or Adams.

2. The best constant $C$ is related to the geometry of $\Omega$ (and also to the eigenvalues of the laplacian operator).