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 \(\varphi\) of \(D\) onto itself with the Jacobian \(h\):

\[\mbox{det}\,\varphi'(x)=h(x). \qquad (1)\] Solving this nonlinear PDE for the components of \(\varphi\) 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 \(t \to \infty\) (the limiting density has to be \(1\) since \([h] = 1\)). The map \(\varphi\), which sends each particle from \(t =0\) to its position at \(t \to \infty\), then satisfies \((1)\).

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

\[\rho_t=\Delta\rho \qquad (2) \] 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

\[\rho\dot{z} = -\nabla\rho; \qquad (3) \] such evolution preserves the mass \(\int\Omega_t\) \(\rho\)dV of any region \(\Omega_t\). Thus, \(h\,dV_0=\rho(x,t)dV_t,\) i.e. \(\frac{dV_t}{dV_0}=\frac{h}{\rho}.\) In the limit \(t \to \infty\) this turns into \((1)\). The “diffusing particle” map \(\varphi\) 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 \(-\mathrm{div}\rho\dot{z}\,dV\stackrel{(3)}{=}\Delta\rho\,dV,\) 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).