Moser’s Theorem on the Jacobians

In one of his seminal papers [1], Moser proved a result, which in the simplest setting, still capturing the gist, states: Given a positive continuous smooth function h on a compact, connected domain  $D\subset R^n$ with the average $[h]=1$, there exists a diffeomorphism $\phi$ of $D$ onto itself with the Jacobian $h$:

$$\text{det}{\phi }^{\prime }(x)=h(x).(1)$$ Solving this nonlinear PDE for the components of $\phi$ may seem like a difficult problem, but a physical analogy leads to a solution at once, as follows.

Interpreting $h$ as the initial density of a chemical dissolved in a medium occupying the domain $D$, we imagine that the chemical diffuses, equalizing its density as $t\to \mathrm{\infty }$ (the limiting density has to be $1$ since $[h]=1$). The map $\phi$, which sends each particle from $t=0$ to its position at $t\to \mathrm{\infty }$, then satisfies $(1)$.

In a bit more detail, let the density $\rho =\rho (x,t)$ evolve according to the heat equation

$${\rho }_t=\mathrm{\Delta }\rho (2)$$ with Neumann boundary conditions (no diffusion through ∂D), starting with $r(x,0)=h(x)$. Assume that each particle $z=z(t)$ diffuses according to

$$\rho \dot{z}=-\mathrm{\nabla }\rho ;(3)$$ such evolution preserves the mass $\int {\mathrm{\Omega }}_t$ $\rho$dV of any region ${\mathrm{\Omega }}_t$. Thus, $hd{V}_0=\rho (x,t)d{V}_t,$ i.e. $\frac{d{V}_t}{d{V}_0}=\frac{h}{\rho }.$ In the limit $t\to \mathrm{\infty }$ this turns into $(1)$. The “diffusing particle” map $\phi$ solves the nonlinear PDE¹. The missing details of this proof are not hard to fill in, or to find in [2].

There has been a lot of work on this problem since Moser’s original paper, in particular on the regularity (references can be found in, e.g., [3]), but my modest goal here was to give a simple basic idea rather than a review of the latest results.

¹ Indeed, the mass enters an infinitesimal patch $dV$ at the rate $-\mathrm{div}\rho \dot{z}dV\stackrel{(3)}{=}\mathrm{\Delta }\rho dV,$ precisely in agreement with $(2)$. Formally, differentiating the mass integral gives two terms which cancel each other. 

Acknowledgments: The work from which these columns are drawn is funded by NSF grant DMS-1412542.

References

[1] Moser, J. On  the  volume  elements  on  a  manifold, Trans. Amer. Math. Soc. 120, 286-294 (1965). 
[2] Levi, M. On a problem by Arnold on periodic motions in magnetic fields, Comm. Pure and Applied Mathematics. 56 (8), 1165-1177 (2003).
[3] Dacorogna, B and Moser, J. On a partial differential equation involving the Jacobian determinant. Ann. l’inst. H. Poincaré Anal. non linéaire. 7(1), 1-26 (1990).