the preimage side and changes of signe, we recognize the graph of $\kappa _T$.

Now we have the following easily verified fact:

\[ir3\] Let $f(x,y,\theta )\in C^\infty (\mathrm{neigh\,}(x_0,y_0,\theta
_0),{\bf R}^n\times {\bf R}^n\times {\bf R}^N)$ be a nondegenerate phase function with $(x_0,y_0,\theta _0)\in C_\phi $, generating a canonical transformation which maps $(y_0,\eta _0)=(y_0,-\nabla
_yf(x_0,y_0,\theta _0))$ to $(x_0,\nabla _xf(x_0,y_0,\theta _0))$. If $g(y)$ is smooth near $y_0$ with $\nabla g(y_0)=\eta _0$ and $$h(x)=\mathrm{v.c.}_{y,\theta }f(x,y,\theta )+g(y)$$ is well-defined with a nondegenerate critical point close to $(y_0,\theta _0)$ for $x$ close to $x_0$, then we have the inversion formula, $$g(y)=\mathrm{v.c.}_{x,\theta }-f(x,y,\theta )+h(x),$$ for $y\in \mathrm{neigh\,}(y_0)$, where the critical point is nondegenerate and close to $(x_0,\theta _0)$.

Combining the three propositions, we get

\[ir4\] [$$\label{ir.6}
G(y,\eta )=\mathrm{v.c.}_{x,\theta }\Im \phi (x,y+i\theta )+\eta \cdot
\theta +\Phi _G(x).$$]{}

If $(\widetilde{\Phi
},\widetilde{G})$ is a second pair of functions close to $\Phi _0$, $0$ and related through (\[ir.2\]), (\[ir.6\]), then [$$\label{ir.7}G\le \widetilde{G}\hbox{ iff }\Phi \le \widetilde{\Phi }.$$]{} Indeed, if for instance $\Phi \le \widetilde{\Phi }$, introduce $\Phi _t=t\widetilde{\Phi }+(1-t)\Phi $, so that $\partial _t\Phi _t\ge 0$. If $G_t$ is the corresponding critical value as in (\[ir.6\]), then $\partial _tG_t=(\partial _t\Phi _t)(x_t)\ge 0$, where $(x_t,\theta _t)$ is the critical point.

Evolution equations 