Let
- $X,Y$ be $\mathbb R$-Hilbert spaces
- $M\subseteq X$ be open
- $\Phi\in C^1(M,Y)$
- $M_0:=\{\Phi=0\}$
- $u\in M_0$ such that ${\rm D}\Phi(u)$ is surjective
- $X_0:=\ker{\rm D}\Phi(u)$
- $X_1:=X_0^\perp$
Let $x_0\in X_0$. How can we show that there is a $\varepsilon_0>0$ and a $\gamma\in C^1((-\varepsilon_0,\varepsilon_0),M_0)$ with $\gamma(0)=u$ and $\gamma'(0)=x_0$?
The idea is as follows: Let $$\Psi(x_0,x_1):=\Phi(u+x_0+x_1)\;\;\;\text{for }(x_0,x_1)\in X_0\times X_1.$$ Since $u\in M$ and $M$ is open, there is a $\varepsilon>0$ with $B_\varepsilon(u)\subseteq M$. Now let $N:=B_{\varepsilon/2}(u)$. Then $\Psi$ is continuously differentiable on $N\times N$. Moreover, $${\rm D}_{x_1}\Psi(0,0)=\left.{\rm D}\Phi(u)\right|_{X_1}$$ is invertible and bounded (by the bounded inverse theorem). Now the implicit function theorem should yield that are open neighborhoods $\Omega_0$ of $0$ in $X_0$ and $\Omega_2$ of $0$ in $X_1$ such that $\Omega_0\times\Omega_1\subseteq N\times N$ and a differentiable $g:\Omega_0\to\Omega_1$ with $$\Psi(x,g(x))=\Phi(u)\;\;\;\text{for all }x\in\Omega_0\tag1.$$ Moreover, $${\rm D}g(x)=-\left({\rm D}_{x_1}\Psi(x,g(x))\right)^{-1}\circ{\rm D}_{x_0}\Phi(x,g(x))\;\;\;\text{for all }x\in\Omega_0\tag2.$$
How can we conclude?
EDIT: I'm really rusty at this stuff, but if I remember correctly, it's easy to construct such a $\gamma$ when $M_0$ is a $k$-dimensional submanifold of $\mathbb R^d$, $k<d$. Surely, that doesn't help here, but I guess these things are related.
I don't think that it helps, but in the problem I'm actually trying to solve, $u$ is an extremal poinf of some $E\in C^1(M)$.