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# 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).

Mark Levi (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.