Geometry of Partial Differential Equations

Chapter 3

Geometry of partial differential equations

In this chapter we introduce the two main classes of partial differential equations that are the subject of this dissertation. The classes are the determined ﬁrst order systems of partial differential equations for two unknown functions of two variables and the second order scalar partial differential equations in the plane. These two classes of equations have a very similar geometric structure and this is precisely the reason that we can develop the theory for these systems in parallel. Small variations of these systems, such as introducing more dependent or independent variables, considering under- or overdetermined systems or higher order equations can lead to systems of partial differential equations for which the geometry is completely different from the geometry in the two systems mentioned above. This makes it difﬁcult to apply the theory to be developed in this dissertation to these other types of systems.

3.1 Ordinary differential equations

We start with describing the geometry of ordinary differential equations to show the methods used. Consider the ordinary differential equation

z00 = F(x, z, z0). (3.1)

The solutions to this equation are functions z(x) that satisfy z00(x) = F(x, z(x), z0(x)). The 2 graph of a 2-jet of a solution is a submanifold of the second order jet bundle J (R). On 2 1 J (R) we have coordinates x, z, p = zx , h = zxx and the two contact forms θ = dz − pdx, θ 2 = dp − hdx. To each function f we associate the submanifold

0 00 2 2 S f = { (x, f (x), f (x), f (x)) ∈ J (R ) | x ∈ R }.

1 2 The submanifolds S f are integral manifolds of the Pfafﬁan system I = span(θ , θ ). 78 Geometry of partial differential equations

If f is a solution to the differential equation (3.1), then S f is a submanifold of the hyper- 2 surface M ⊂ J (R) defined by h = F(x, z, p). Conversely, every integral manifold of the Pfaffian system I restricted to M for which dx 6= 0 is locally the graph of a solution. The correspondence between solutions of (partial) differential equations and integral manifolds of a Pfaffian system (or more general an exterior differential ideal) will be used many times.

3.2 Second order scalar equations

We consider a second order partial differential equation for the unknown function z and the independent variables x, y. We introduce the coordinates x, y, z, p = zx , q = zy, r = zxx , 2 2 s = zxy, t = zyy for the second order contact bundle J (R , R). The use of p, q and r, s, t for the ﬁrst and second order derivatives, respectively, was introduced by Monge. The coordinates x, y, z, p, q, r, s, t are called Monge coordinates or classical coordinates. The most general form of such an equation is given by F(x, y, z, p, q, r, s, t) = 0. (3.2)

We require that that (Fr , Fs, Ft ) 6= 0, so that (3.2) deﬁnes a truly second order equation. If Fr 6= 0, then we can (locally) solve for r and rewrite the equation as r = ρ(x, y, z, p, q, s, t). (3.3)

If Fr = 0 at a point, then we can either solve for one of the other second order variables or we can make a coordinate transformation x 7→ x + y, y 7→ x − y or x 7→ y, y 7→ x such that in the new coordinates Fr 6= 0. Let us analyze the geometry of such an equation. On the second order jet bundle the equation (3.2) or (3.3) deﬁnes a hypersurface M. On the second order contact bundle we have the contact forms θ 0 = dz − pdx − qdy, (3.4) θ 1 = dp − rdx − sdy, θ 2 = dq − sdx − tdy.

These contact forms pull back to contact forms on M, which will also be denoted by θ j . Assume that Fr 6= 0 and that we can solve for r as r = ρ(x, y, z, p, q, s, t). On the hypersurface M the contact forms are θ 0 = dz − pdx − qdy, (3.5) θ 1 = dp − ρdx − sdy, θ 2 = dq − sdx − tdy. The solutions of the partial differential equation (3.2) are locally in one-to-one correspondence with the integral manifolds of the Pfafﬁan system generated by the forms (3.5) that satisfy the independence condition dx ∧ dy 6= 0. The distribution V dual to the contact forms is spanned by

∂x + p∂z + ρ∂p + s∂q , ∂y + q∂z + s∂p + t∂q , (3.6) ∂s, ∂t . 3.3 First order systems 79

A simple calculation shows that 0 C(V) = span ( 0 ) , V = span V, ∂p, ∂q , 0 00 0 C(V ) = span (∂s, ∂t ) , V = span V , ∂z = TM. These expressions are only expressions in local coordinates, but since ρ was general we can see that all systems that come from second order scalar equations have a rank 4 distribution V with the properties that rank V0 = 6, rank V00 = 7, rank C(V0) = 2, C(V0) ⊂ V. These conditions on V describe the entire geometry of the equation. We will see in Section 4.1 that distributions with these properties can locally be written as a second order equation.

3.3 First order systems

Consider a ﬁrst order system of partial differential equations in the independent variables x1,..., xn and dependent variables u1,..., us deﬁned by the equations Fλ(x, u, p) = 0 λ = 1,..., c. (3.7)

i i j Here p j stands for the first order derivative ∂u /∂x . We can define this system in a co- n s ordinate invariant way as follows. Let B be an open subset of R × R and consider the Grassmann bundle Grn(TB) of n-planes over B. If we assume the equations (3.7) are of constant maximal rank c and we can solve for the first order variables p, then the equations define a codimension c submanifold M of the Grassmann bundle and the canonical projection π : M → B is a submersion. On the Grassmann bundle Gr2(TB) we have the contact system i = i − i j = I , which is generated by the contact forms θ du p j dx , i 1,..., s. The pullbacks of the contact forms define the contact system on M. This contact system is of constant rank and defines the contact distribution V = I ⊥. Solutions u(x) of the system (3.7) correspond to integral manifolds of the contact system on M. We will specialize to the case n = s = c = 2. For such first order systems we will write x, y for the independent variables, u, v for the dependent variables and p = ux , q = u y, r = vx , s = vy for the first order derivatives. The equation manifold M has dimension 6. From this point on we adopt the convention that, unless stated otherwise, a first order system is a first order system of two partial differential equations in two independent and two dependent variables. The definition of such a system can be given by equations in local coordinates, in terms of a distribution on a 6-dimensional manifold or in terms of a codimension 2 Pfaffian system. Any codimension 2 first order system is defined by two equations F1 = 0, F2 = 0 and can be characterized as follows. If we define ω1 = dx, ω2 = dy, then the structure equations for the contact forms are 1 1 π1 π2 dθ ≡ 2 2 ∧ ω mod I. π1 π2 Notice that any system satisfies dθ ≡ 0 mod J, J = span(I, ω1, ω2). Hence the graphs of solutions to the partial differential equation correspond to integral manifolds of the linear 80 Geometry of partial differential equations

Pfaffian system (I, J). Since dω ≡ 0 mod J, the distribution dual to J is an integrable rank two distribution. The leaves of this distribution are precisely the fibers of the projection M → B. The contact distribution V on M is the distribution dual to I . The Lie brackets modulo the subbundle define a tensor V ×M V → TM/V. We will see later that for elliptic and hyperbolic systems this tensor is non-degenerate (the precise definition of non-degenerate is given in Appendix A.1). We will use these properties to define a generalization of a first order system in which there is no distinguished set of independent or dependent variables.

Definition 3.3.1. A generalized first order system under contact geometry is a smooth manifold M of dimension 6 with a rank 4 distribution V such that the Lie brackets modulo the subbundle is a non-degenerate map V ×M V → TM/V.A generalized first order system under point geometry is a generalized first order system (M, V) with an integrable distribution U ⊂ V.

A contact transformation of a first order system is a diffeomorphism leaving invariant V. A point transformation is a diffeomorphism leaving invariant both V and U. The discussion above shows that any elliptic or hyperbolic first order system defines a generalized first order system under contact geometry and a generalized first order system under point geometry as well. In Section 4.6.1 we will see that any analytic generalized first order system can be written locally as an elliptic or hyperbolic system of partial differential equations. The parabolic first order systems can also be formulated on a 6-dimensional manifold with rank 4 distribution. The author is not aware of conditions on V that guarantee that (M, V) is locally equivalent to a first order parabolic system. For some non-generic distributions (for example if the distribution is integrable), there are no corresponding systems of equations. The definition (3.3.1) of a generalized first order system under point geometry is very similar to the definition in McKay [51, p. 37] of almost generalized Cauchy-Riemann equations. The only differences are that McKay only defines the systems for equations of elliptic type (for the type of first order system see Section 4.6) and that he introduces an orientation for the manifold M.

Remark 3.3.2. A structure theory of first order systems in terms of vector fields was already introduced by Vessiot [67]. Dual to the formulation in vector fields there is a formulation in differential forms. For first order elliptic systems McKay [51] has developed the structure theory in great detail. The structure theory using differential forms and vector fields is developed by Vassiliou for hyperbolic systems [65, 66].