Consider the following functional:
$\;E(u)=\int_{\mathbb R} \frac{1}{2}{\vert u' \vert}^2+W(u)\;dx\;$ where $\;u \in C^4(\mathbb R;\mathbb R^2)\;,\;W \in C^3(\mathbb R^2;\mathbb R)\;$
- Denote $\;S:=\{$the set in which the energy functional is minimized$\}\;$ and assume the difference of elements of $\;S\;$ belongs to $\;W^{1,2}(\mathbb R;\mathbb R^2)\;$
- Let $\;u \in S\;$ and $\;F \subset S\;$, the the distance from $\;u\;$ to $\;F\;$ in the $\;W^{1,2}-$norm is: $\;d(u,F)=inf\{{\vert \vert u-v \vert \vert}_{W^{1,2}}: v\in F\}\;$
- Call $\;\mathcal Z\;$ the set of all minimizers of $\;E\;$ over $\;S\;$ and $\;C(z)=\{z(\cdot-m):m\in \mathbb R\;,\;z\in \mathcal Z\}\;$
- $\;e_1=\min\{E(u):u\in S\}\;$
If there are strictly positive numbers $\;a,b,c\;$ such that:
- $\;\forall a\gt 0\;\Rightarrow \inf\{E(u):u\in S\;,\;d(u,\mathcal Z)\ge a\}\gt e_1\;$
- if $\;d(u,\mathcal Z \setminus C(z))\gt 0\;$ and $\;d(u,C(z)) \lt b \Rightarrow E(u)-e_1\ge c{d(u,C(z))}^2\;$
Prove that $\;\forall a\in(0,d(z,\mathcal Z \setminus C(z))\;$ there exists $\;\beta \gt 0\;$ such that:
$\;d(u,\mathcal Z \setminus C(z)) \ge a \Rightarrow E(u)-e_1 \ge \beta \min\{1,{d(u,C(z))}^2\}\;$
Our professor said that this kind of exercise is quite easy since it follows immediately from the two last inequalities. However I'm having a really hard time solving this.
My Attempt:
$\;E(u)-e_1 \ge \beta \min\{1,{d(u,C(z))}^2\}\;$ implies that I should have had $\;E(u)-e_1 \ge \beta\;$ and $\;E(u)-e_1 \ge \beta {d(u,C(z))}^2\;$. The second inequality is given from the data. Now for the first one, I believe since $\;\inf\{E(u):u\in S\;,\;d(u,\mathcal Z)\ge a\}\gt e_1\;$ it follows $\;E(u) \ge e_1+\beta\;$ for some $\;\beta \gt 0\;$
Moreover in order for all the above inequalities to hold, it should $\;d(u,\mathcal Z\setminus C(z))\ge a\;$ since $\;\mathcal Z\setminus C(z) \subset \mathcal Z \Rightarrow d(u,\mathcal Z\setminus C(z)) \ge d(u,\mathcal Z) \ge a\;$
Still, I'm a bit unsure if the above thoughts are valid. I would appreciate any help!
Thanks in advance