Lagrangian Solutions
to
Systems of
Real Principal Type
Dissertation
zur Erlangung des Doktorgrades
des Fachbereichs Mathematik-Informatik
der Universit¨
at Paderborn
vorgelegt von Daniel R¨
ohrig
Paderborn, Dezember 2001
Gutachter: Prof. Dr. S¨onke Hansen
Prof. Dr. Gunther Uhlmann
Prof. Dr. Hermann Sohr
Tag der Promotion: 14. Februar 2002
Contents
1. Introduction 1
2. Preliminaries 4
2.1. Systems of Pseudodifferential Operators 4
2.2. Lagrangian Distributions 8
2.3. Systems of Real-Principal Type 11
3. Lagrangian Solutions 17
3.1. Statement of the Results 17
3.2. Special Inhomogeneous Equations. 19
3.3. Proof of the Theorems 24
4. Application to the Elastodynamics Equation 26
References 34
1
1. Introduction
We consider the wave equation
v=c−2(x)∂2
tv−∆v=0,
on Rt×R3
x. This is the prototype of the class of hyperbolic operators,
which describe wave-like propagation phenomena.
To find solutions to the wave equation, one can try the classical
ansatz of geometrical optics: Consider a function
v(t, x)=a(x, ω)eiω(φ(x)−t),a(x, ω)=
∞
k=0
(iω)−kak(x).(1.1)
with amplitude a. Here the principal part a0of the amplitude should
be unequal to 0.
By inserting (1.1) into the wave equation, an elementary calcula-
tion yields the following two conditions: φneeds to solve the eikonal
equation,
c|∇φ|=1,
and a0needs to solve the transport equation,
0=2(∇φ·∇)a0+∆φa
0,
here ∇is the formal differential operator (∂x1,∂
x2,∂
x3); ∇φis the gra-
dient of λ,∇φ·∇ is the derivation in ∇φ-direction and so forth.
If these two equations are satisfied, one has that v=O(ω2−2)=
O(ω0)forω→∞.
The linear transport equation can be solved by reduction to ordinary
differential equations along rays, which are the orthogonal trajectories
of the wavefronts φ= constant. By iterating this method with special
inhomogeneous equations, one gets an asymptotic solution vsuch that
for all k∈N
v=O(ω2−k)forω→∞.(1.2)
In general, the ansatz of geometrical optics does not provide global
solutions. In so-called caustics this method breaks down, cf. for exam-
ple Duistermaat [2, section 5.2].
The ansatz (1.1) translates into special Lagrangian distributions u∈
I0(X,Λ): Here X:= Rt×R3
xand uis of the form
u(t, x)=(2π)−3/2eiω(ϕ(x)−t)a(x, ω)dω.
Λ is the Lagrangian submanifold
{(t, x, τ, ξ)∈TX\0|t=ϕ(x),τ=−ωand ξ=−τ∇φ(x)},
Loading more pages...