**Theorem** Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with Lipschitz boundary. Let $p$ be a continuous semi-norm on $H^1(\Omega)$, and $p$ is a norm on the constants (i.e. That $p(u)=0$ and $u$ is a constants imply that $u=0$). Then there exists a constant $c$ depending only on $\Omega$ such that $$(*)\hspace{25pt}\norm{ u }_{L^2} \leq c(\Omega)( \norm{\grad u}_{L^2} + p(u) ),$$ holds for any $u\in H^1(\Omega)$. //Proof// We prove the theorem by contradiction. Suppose for each $n$, there exists $u_n\in H^1(\Omega)$ such that $$(1)\hspace{25pt}\norm{u_n}_{L^2} > n(\norm{\grad u_n}_{L^2} + p(u_n)).$$ By scaling, we may assume that $\norm{u_n}_{L^2} = 1$ for all $n$. By (1), we then find $$ \norm{\grad u_n}_{L^2} + p(u_n) \leq \f{1}{n}\rightarrow 0,\quad (n\rightarrow \infty).$$ Therefore, we find $$\grad u_n \rightarrow 0,\quad p(u_n)\rightarrow 0,\quad (n\rightarrow \infty),$$ which implies that $$u_n\rightarrow u_0=constant,$$ together with $p$ is a continuous semi-norm and is a norm on the constant, we find that $$u_0=0.$$ That contradicts with the assumption that $\norm{u_n}_{L^2}=1$ for all $n$. Hence, there exists a constant $c(\Omega)$ satisfying (*). === Remark #The idea of this proof comes from $\Gamma e\Pi b\phi a h \pi$ Lemma (see [张林87, pp. 105] #The Hilbert space $H^1(\Omega)$ can be replaced by general Sobolev space $W^{1,p}(\Omega)$. #In this proof, the boundedness of the domain $\Omega$ is necessary since we need the constant (function) belongs to $H^1(\Omega)$. #There is a version "Poincare inequality" for the sets bounded in one direction, see [Tem01, pp. 9]. === References #[Tem97]R. Temam, Infinite Dimensonal Dynamical Systems in Mechanics and Physics, 2nd Edition, Springer-Verlag, New York, 1997 #[Tem01]R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. #[张林87]张恭庆, 林源渠, 泛函分析讲义上册, 北京大学出版社,1987.