PomeProposition 14.4.1.2AD3CD874EA4 With this linear normalization, the solution $(u,b)$ of (14.1.1) depends smoothly on $F$ in Hölder spaces, and hence smoothly on the parameters of every smooth finite-dimensional family of smooth data.
14.4 The linearized equation and smooth dependence7BDF4C42EB78
7BDF4C42EB78
Recall from (11.23.1) that the linearization of $u\mapsto \log (\omega _ u^ n/\mathrm{d}V)$ at $u$ is $L_ uv=n\, \mathrm{dd}^{\mathrm{c}}v\wedge \omega _ u^{n-1}/\omega _ u^ n$. This operator annihilates constants, and by proposition 11.23.2 its range consists of the functions with zero mean under the adjoint probability measure $\mu _ u$. Including the scalar $b$ as an unknown removes this obstruction. In dimension one, $\mu _ u=\omega _ u/\int _ X\omega _ u$, and the compatibility condition follows from Stokes' theorem.
The normalization $\sup _ Xu=0$ is convenient for compactness, but it need not depend differentiably on the data. To differentiate solutions, we use instead the linear normalization
\[ \int _ Xu\, \mathrm{d}V=0. \]
Passing from one normalization to the other only adds a constant to $u$ and leaves $b$ unchanged.
Proof. Fix an integer $k\geq 0$ and a real exponent $0<\alpha <1$. Consider the map
\[ (u,b,F)\longmapsto \log \frac{(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n}{\mathrm{d}V}-F-b \]
on the open set of potentials in $C^{k+2,\alpha }$ with $\omega _ u>0$, restricted to $\int _ Xu\, \mathrm{d}V=0$. Its derivative in $(u,b)$ is
\[ (v,a)\longmapsto L_ uv-a. \]
Given $h\in C^{k,\alpha }$, integrating $L_ uv-a=h$ against $\mu _ u$ forces $a=-\int _ Xh\, \mathrm{d}\mu _ u$. For this value, the equation for $v$ satisfies the compatibility condition of proposition 11.23.2, and it has a unique solution with zero $\mathrm{d}V$-mean. Schauder estimates show that this inverse is bounded. The implicit function theorem in Banach spaces therefore gives a smooth local solution map around any given solution; this step does not use the global existence theorem. Uniqueness identifies the local maps wherever they overlap, and elliptic regularity allows arbitrary $k$ for smooth data. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.