By Mark Levi
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 ([email protected]) is a professor of mathematics at the Pennsylvania State University.