I'm reading a paper in Calculus of Variation and I've been stuck to the proof of the following Corollary. But first let me introduce the problem:
NOTE: There is a typo in the proof of Corollary 3.2. There is no Lemma 3.2 in the paper. It means the lower bound in the statement of Corollary 3.2.
What I cannot see that easy is why $\;\inf \{E_1(u):u\in S(a,b),d_1(u,\mathcal Z) \ge \beta\} \lt e_1\;$ is impossible and hence the proof is complete after contradicting only the case of equality with $\;e_1\;$. I do understand that in case the equality holds , we lead to a contradiction but I don't see why the writer doesn't examine also the case in which $\;\inf \{E_1(u):u\in S(a,b),d_1(u,\mathcal Z) \ge \beta\} \lt e_1\;$..
Any help would be valuable! Thanks in advance
It is because of your definition of the minimzer $e_1$ by: \begin{align} e_1 = \min\{ E_1(v): v \in S(a,b) \} \end{align} Now if your assume that in the above corollar the equality holds than, by the property of the infimum you can find a sequence $u_n$ so that: \begin{align} E_1(u_n) \longrightarrow e_1 = E_1(z) \quad (n \rightarrow \infty) \text{ and $d(u_n,z) \geq \beta$ for some $z \in \mathcal{Z}$ } \end{align} If you find a sequence $x_n$, which converges pointwise to $z$ then the distance to $\mathcal{Z}$ tends especially to zero, because we have taken $z \in \mathcal{Z}$.