Let us consider a manifold ${M}$ and a hypersurface ${N \subset M}$. Given a smooth function ${F: T^*M \rightarrow \mathbb{R}}$, we are interested in solving for ${u \in C^{\infty}(M)}$ the eikonal equation

$\displaystyle \boxed{ F( du ) = 0 \ (*)}$

where the initial condition is ${u|_N = s}$ for ${s: N \rightarrow \mathbb{R}}$ smooth.

(The phrase “eikonal equation” seems to refer to a specific case of this, but I’ll follow Taylor in applying the term to this more general situation.)

Using the story already discussed about symplectic forms, we can do this locally in the noncharacteristic case.

To define this, we pick local coordinates ${x^1,\dots, x^n}$ around a fixed ${p_0 \in N}$ such that ${N}$ is given by ${x^n = 0}$. Let the corresponding local coordinates for ${T^*M}$ be ${x^1, \dots, x^n}$ together with ${\xi^1, \dots, \xi^n}$. Choose a cotangent vector ${v = \sum v_i dx^i \in T_{p_0}^*(M)}$ such that ${(ds)_{p_0} = \sum_{i < n} v_i dx^i}$. Suppose ${F(v)=0}$, and ${\frac{\partial F}{\partial \xi_n} \neq 0}$. This is the noncharacteristic hypothesis.

Theorem 1 Under the noncharacteristic hypothesis, the eikonal equation can be locally solved—that is, there exists a neighborhood ${U}$ containing ${p_0}$ and a smooth function ${u}$ on ${U}$ satisfying (*) with ${u|_{U \cap N} = s}$.  Moreover, $du|_{p_0} = v$.

The idea of the proof of this existence theorem is to construct ${u}$ by constructing ${du}$, which in turn is done by constructing its graph—which after all must satisfy a specific equation—as a submanifold of ${T^*M}$. At first it will not be obvious that the graph actually corresponds to any ${du}$. This will follow from the analysis of lagrangian submanifolds.

Lagrangian submanifolds of the cotangent bundle

Given a symplectic vector space ${V}$ (i.e. a vector space equipped with a nondegenerate skew-symmetric form ${\omega}$), a subspace ${W \subset V}$ is said to be lagrangian if ${\dim W = \frac{1}{2} \dim V}$ and ${\omega|_W \equiv 0}$. There is an easy example if we are given a symplectic basis ${e_1, \dots, e_n, f_1, \dots, f_n}$ with ${\omega(e_i,e_j)=0, \omega(f_i, f_j)=0, \omega(e_i, f_j) = \delta_{ij}}$. Then the subspaces ${W ,W'}$ spanned by the ${e_i}$ (resp. ${f_i}$) are lagrangian.

A submanifold ${N}$ of a symplectic manifold ${M}$ is lagrangian iff for ${n \in N}$, ${T_n(N) \subset T_n(M)}$ is a lagrangian subspace.

Proposition 2 Let ${t: M \rightarrow T^*M}$ be a section (i.e. 1-form) embedding ${M}$ as a submanifold of the cotangent bundle, which is endowed with the canonical form. Then ${t(M)}$ is lagrangian iff ${t}$ is closed.

We will prove this by using local coordinates ${x^1, \dots, x^n}$ for ${M}$ and corresponding ${\xi^1, \dots, \xi^n}$ which together form a system of local coordinates on ${T^*M}$. The symplectic form on ${T^*M}$ is then locally

$\displaystyle \omega = \sum dx^i \wedge d\xi^i.$

Suppose in this local coordinates the map ${t}$ is defined by

$\displaystyle t(x^1, \dots, x^n) = ( \Xi^1, \dots, \Xi^n)$

or in other words

$\displaystyle t = \sum \Xi^i dx^i .$

Then a basis for the tangent space of ${t(M)}$ is given by

$\displaystyle \left\{ t_i = dx^i + \sum_j \frac{\partial \Xi^j}{\partial x^i} d\xi^j \ 1 \leq i \leq n \right\}$

and we can compute

$\displaystyle 0 = \omega( t_i, t_j) = \frac{\partial \Xi^j}{\partial x^i} - \frac{\partial \Xi^i}{\partial x^j}$

Comparing this with the local expression for ${dt}$ shows that this is precisely the closedness condition.

Incidentally, I’m curious if there is an invariant proof here. Any ideas?

Constructing a surface in the cotangent bundle

We are now going to define the surface that will be the graph of ${du}$, where ${u}$ is the solution of the eikonal equation (*). We start with the fiber over ${N}$. Let ${i: N \rightarrow M}$ be the inclusion.

Define

$\displaystyle R' := \{ w \in T_n(M): n \in N, i^*w = (ds)_n, F(w) = 0.$

In words, ${R'}$ consists of cotangent vectors lying over ${N}$ that annihilate ${F}$ and which correspond to the differential of the boundary condition ${s}$ plus possibly something extra in another direction. Near ${v}$, ${R'}$ is a surface of dimension ${n-1}$ by the implicit function theorem and the noncharacteristic hypothesis. Take a neighborhood and intersect it with ${R'}$ to get ${R}$. So ${R'}$ is a surface over the intersection of ${N}$ with some neighborhood of ${p}$ then.

However, we need to extend this surface over a neighborhood of ${p}$ in ${M}$. We want ${F}$ to be zero on this surface, and we know that ${F}$ is constant on the integral curves of the Hamiltonian vector field ${H_F}$ because ${H_F F = 0}$. So, we take the union of ${R}$ with the integral curves of ${H_F}$ through points of ${R}$. I now claim that this is a surface indeed. As before, choose local coordinates ${x^1, \dots, x^n}$ about ${p}$ and let ${\{x^1, \dots, x^n, \xi^1, \dots, \xi^n\}}$ be the local coordinates for ${T^*M}$. The symplectic form is locally given as ${\sum dx^i \wedge d \xi^i}$. Then because ${\frac{\partial F}{\partial \xi_n} \neq 0}$, ${H_F}$ includes a nonzero multiple of ${\frac{\partial}{\partial x_i}}$. So this is indeed a surface of dimension ${n}$, which can be represented in some neighborhood of ${p}$ as a section of ${M}$ into ${T^*M}$. And on this surface, ${F \equiv 0}$.

I claim now that the surface ${S}$ thus constructed is lagrangian, so we will be able to apply the previous result. So if ${q \in S}$ and ${a,b \in T_q(S)}$, we must prove then ${\omega(a,b)=0}$.

We first do this when ${q \in R}$. If we actually have tangency to ${R}$, not simply ${S}$, i.e. ${a,b \in T_q(R)}$, then the fact that ${\omega(a,b)=0}$ follows from Proposition 2 with ${N}$ replacing ${M}$ that ${\omega(a,b)=0}$, since the portion of the tangent vectors pointing orthogonally to ${T_q(R)}$ cancel each other out. (I.e. use local coordinates ${x^1, \dots, x^n}$ so ${N}$ is given by ${x^n=0}$; then the portions of ${a,b}$ with ${\frac{\partial}{\partial \xi^n}}$ in them annihilate each other. The other parts, which consist of ${\frac{\partial}{\partial x^i}, \frac{\partial}{\partial \xi^i}}$ for ${i < n}$, do so as well by what has already been proved.) Next, if one of ${a,b}$ is a multiple of ${H_F}$ at the point ${q}$, then

$\displaystyle \omega(a,b) = aF = 0$

where ${a F}$ denotes the application of the tangent vector ${a}$ to ${F}$. If ${a,b}$ are both multiples of ${H_F}$ at ${q}$, then trivially ${\omega(a,b)=0}$. So the symplectic form vanishes on ${T_q(S)}$ if ${q \in R}$. The general case because the flows of the Hamiltonian vector field ${H_F}$ preserve the symplectic form ${\omega}$, and every point on ${S}$ is obtained by such a flow from ${R}$.

So, given the lagrangianness of the surface ${S}$, we have thus found ${u}$ with ${du}$ satisfying the eikonal PDE above. Also, ${d(u-s)_S = 0}$, so by modifying ${u}$ by a constant we get the appropriate boundary condition.

Next, I’ll try to say something about uniqueness, and more generally first-order nonlinear PDEs.  After that, I’m thinking of possibly a post or two on integrable systems and perhaps the Arnold-Liouville theorem.  Then perhaps some tools from analysis (e.g., distributions, Fourier transforms, Sobolev spaces, etc.) to get to the general theory of linear PDEs.