Arguing by contradiction

THEOREM 14.1. Let $k \geqslant 0$ be an integer, and let $p \in[1, \infty]$. There exists a constant $C(\Omega)$ such that
$$
\inf _{p \in P_k(\Omega)}\|v+p\|_{k+1, p, \Omega} \leqslant C(\Omega)|v|_{k+1, p, \Omega} \quad \text { for all } v \in W^{k+1, p}(\Omega)
$$
and consequently, such that
$$
\|\dot{v}\|_{k+1, p, \Omega} \leqslant C(\Omega)|\dot{v}|_{k+1, p, \Omega} \quad \text { for all } \dot{v} \in W^{k+1, p}(\Omega) / P_k(\Omega) \text {. }
$$
Proof. Let $N=\operatorname{dim} P_k(\Omega)$ and let $f_i, 1 \leqslant i \leqslant N$, be a basis of the dual space of $P_k(\Omega)$. We will show that
$$
\|v\|_{k+1, p, \Omega} \leqslant C(\Omega)\left(|v|_{k+1, p, \Omega}+\sum_{i=1}^N\left|f_i(v)\right|\right) \quad \text { for all } v \in W^{k+1, p}(\Omega) \text {. }
$$

Inequality (14.9) will then be a consequence of inequality (14.11): Given any function $v \in W^{k+1, p}(\Omega)$, let $q \in P_k(\Omega)$ be such that $f_i(v+q)=0,1 \leqslant i \leqslant N$. Then by (14.11),
$$
\inf _{p \in P_k(\Omega)}\|v+p\|_{k+1, p, \Omega} \leqslant\|v+q\|_{k+1, p, \Omega} \leqslant C(\Omega)|v|_{k+1, p, \Omega},
$$

which proves (14.9). If inequality (14.11) is false, there exists a sequence $\left(v_l\right)$ of functions $v_l \in W^{k+1, p}(\Omega)$, such that
$$
\begin{aligned}
&\left\|v_l\right\|_{k+1, p, \Omega}=1 \text { for all } l \geqslant 1, \\
&\lim _{l \rightarrow \infty}\left\{\left|v_l\right|_{k+1, p, \Omega}+\sum_{i=1}^N\left|f_i\left(v_l\right)\right|\right\}=0 .
\end{aligned}
$$
Since the sequence $\left(v_l\right)$ is bounded in $W^{k+1, p}(\Omega)$, there exists a subsequence, again denoted $\left(v_l\right)$ for notational convenience, that converges in the space $W^{k, p}(\Omega)$ (this follows from the Kondrasov or Rellich theorems for $1 \leqslant p<\infty$ and from Ascoli's theorem for $p=\infty$ ). Since
$$
\lim _{l \rightarrow \infty}\left|v_l\right|_{k+1, p, \Omega}=0,
$$
by (14.12), and since the space $W^{k+1, p}(\Omega)$ is complete, the sequence $\left(v_l\right)$ converges in the space $W^{k+1, p}(\Omega)$.
The limit $v$ of this sequence is such that
$$
\left|\partial^\alpha v\right|_{0, p, \Omega}=\lim _{l \rightarrow \infty}\left|\partial^\alpha v_l\right|_{0, p, \Omega}=0 \quad \text { for all } \alpha \text { with }|\alpha|=k+1 \text {, }
$$
and thus $\partial^\alpha v=0$ for all multi-indices $\alpha$ with $|\alpha|=k+1$. Since a domain is connected . by assumption, it follows from distribution theory (see SCHWARTZ [1966, p. 60] that the function $v$ is a polynomial of degree $\leqslant k$. Using (14.12), we have
$$
f_i(v)=\lim _{l \rightarrow \infty} f_i\left(v_l\right)=0 ;
$$
hence we conclude that $v=0$ since $v \in P_k(\Omega)$. But this contradicts the equality $\left\|v_l\right\|_{k+1, p, \Omega}=1$ for all $l$.

 

引自：

P.G. Ciarlet, Basic Error Estimates for Elliptic Proplems, in: P.G. Ciarlet, J.L. Lions, (Ed.), Finite Element Methods (Part1), Handbook of Numerical Analysis, vol.2, Elsevier Science Publishers, North-Holand, 1991, pp.21-343.