Inner-variational equations

It’s been a while since the last time I posted a post-colloquium post. This one is based on a colloquium given by me, but don’t worry: none of my results are included here.

As a warm-up, consider the (trivial) problem of finding the shortest path between two points a,b\in\mathbb R^n. The naive approach is to minimize the length L(f)=\int_0^1 |f\,'(t)|\,dt among all maps f\colon [0,1]\to \mathbb R^n that are sufficiently smooth and satisfy the boundary conditions f(0)=a and f(1)=b. This turns out to be a bad idea: L(f) is neither strictly convex nor differentiable, and the set of minimizing maps is huge, containing some rather nonsmooth specimen.

The right approach is to minimize the energy E(f)=\int_0^1 |f\,'(t)|^2\,dt. While this functional is not immediately related to length for general maps, it is not hard to see that for minimizing maps f we have E(f)=L(f)^2. Indeed, consider performing the inner variation \widetilde{f}(t)=f(t+\epsilon \eta(t)) where \eta\colon [0,1]\to\mathbb R is smooth and vanishes at the endpoints. Expanding the inequality E(\widetilde{f})\ge E(f), we arrive at \int_0^1 |f\,'(t)|^2\eta'(t)\,dt=0, which after integration by parts yields \frac{d}{dt}|f\,'(t)|^2=0. Thus, minimization of energy enforces constant-speed parametrization, and since for constant-speed maps we have E(f)=L(f)^2, the geometric nature of the variational problem has not been lost.

As a side remark, the inner-variational equation \frac{d}{dt}|f\,'(t)|^2=0 could be written as f\,''\cdot f\,'=0, which is a nonlinear second-order equation.

For comparison, try the first variation \widetilde{f}(t)=f(t)+\epsilon \eta(t) where \eta\colon [0,1]\to\mathbb R^n is smooth and vanishes at the endpoints. Expanding the inequality E(\widetilde{f})\ge E(f), we arrive at \int_0^1 f\,'(t)\cdot \eta'(t)\,dt=0, which after integration by parts yields f\,''\equiv 0, a linear second-order equation. This Euler-Lagrange equation immediately tells us what the minimizing map f is: there is only one linear map with the given boundary values. Obviously, f\,''\equiv 0 is a much stronger statement that f\,''\cdot f\,'=0. However, if f+\epsilon \eta is not admissible for some geometric reason (e.g., the curve must avoid an obstacle), the Euler-Lagrange equation may not be available.

Moving one dimension up, consider the problem of parameterizing a given simply-connected domain \Omega\subset \mathbb C. Now we are to minimize the energy of diffeomorphisms f\colon \mathbb D\to\Omega which is defined, as before, to be the sum of squares of derivatives. (Here \mathbb D is the unit disk.) In complex notation, E(f)=\iint_{\mathbb D}(|f_z|^2+|f_{\bar z}|^2). For definiteness assume f is sense-preserving, that is |f_z|\ge |f_{\bar z}|. Minimizers ought to be conformal maps onto \Omega, but how can we see this from variational equations?

The Euler-Lagrange equation that we get from E(f)\le E(f+\epsilon\eta) turns out to be the Laplace equation \Delta f=0. This is much weaker than the Cauchy-Riemann equation f_{\bar z}=0 that we expect. One problem is that \eta must vanish on the boundary: otherwise f+\epsilon\eta will violate the geometric constraint. We could try to move the values of f in the direction tangent to \partial\Omega, but since does not necessarily make sense since the boundary of \Omega could be something like the von Koch snowflake. And of course, it is not at all clear why the minimum of E must be attained by a diffeomorphism. If the class of maps is expanded to include suitable limits of diffeomorphisms, then it’s no longer clear (actually, not true) that f+\epsilon\eta belongs to the same class. All things considered, the approach via the first variation does not appear promising.

Let’s try the inner variation instead. For small \epsilon the map z\mapsto z+\epsilon\eta is a diffeomorphism of \mathbb D, hence its composition with f is as good a candidate as f itself. Furthermore, since the inner variation deals with the model domain \mathbb D and not with the generic domain \Omega, it is easy to allow modification of boundary values: \eta should be tangent to \partial \mathbb D, i.e., \mathrm{Re}\,\bar z\eta(z) should vanish on the boundary. It takes a bit of computation to turn E(f(z+\epsilon \eta(z))) into something manageable, but the final outcome is remarkably simple. The inner-variational equation says that the function \varphi:=f_z\overline{f_{\bar z}} is holomorphic in \mathbb D and z^2 \varphi is real on the boundary \partial \mathbb D. (Technically speaking, \varphi\, dz^2 must be a real holomorphic quadratic differential.) What can we conclude from this? To begin with, the maximum principle implies that z^2 \varphi is a constant function. And since it vanishes at z=0, the inevitable conclusion is \varphi\equiv 0. Recalling the sense-preserving constraint |f_z|\ge |f_{\bar z}|, we arrive at f_{\bar z}\equiv 0, the desired Cauchy-Riemann equation.

Executive summary

  • Suppose we are to minimize some quantity (called “energy”) that depends on function (or a map) f
  • We can consider applying the first variation f+ \epsilon\eta of the inner variation f\circ (\mathrm{id}+\epsilon \eta). Here \eta is a small perturbation which is applied differently: in the first case it changes the values of f, in the second it shuffles them around.
  • Inner variation applies even in the presence of geometric constraints that make first variation illegal. One example of such constraint is “f must be injective”.
  • Inner-variational equations are quite different from the Euler-Lagrange equations. Even for simple quadratic functionals they are nonlinear.
  • Inner-variational equations are useful because they tell us something about the maps of minimal energy.

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