Tiểu Luận Propagation of singularities for the wave equation on manifolds with corners

Propagation of singularities for the wave
equation on manifolds with corners
By Andr as Vasy*
Abstract
In this paper we describe the propagation of C
1
and Sobolev singularities
for the wave equation on C
1
manifolds with corners M equipped with a Rie-mannian metric g. That is, for X = M  Rt
, P = D
2
t

M , and u 2 H
1
loc
(X)
solving Pu = 0 with homogeneous Dirichlet or Neumann boundary condi-tions, we show that WF
b
(u) is a union of maximally extended generalized
broken bicharacteristics. This result is a C
1
counterpart of Lebeau's results
for the propagation of analytic singularities on real analytic manifolds with
appropriately stratied boundary, [11]. Our methods rely on b-microlocal pos-itive commutator estimates, thus providing a new proof for the propagation of
singularities at hyperbolic points even if M has a smooth boundary (and no
corners).
1. Introduction
In this paper we describe the propagation of C
1
and Sobolev singularities
for the wave equation on a manifold with corners M equipped with a smooth
Riemannian metric g. We rst recall the basic denitions from [12], and refer
to [20, x2] as a more accessible reference. Thus, a tied (or t-) manifold with
corners X of dimension n is a paracompact Hausdor topological space with
a C
1
structure with corners. The latter simply means that the local coordi-nate charts map into [0; 1)
k
 R
n k
rather than into R
n
. Here k varies with
the coordinate chart. We write @
`X for the set of points p 2 X such that in
any local coordinates  = (
1
; : : : ; 
k
; 
k+1
; : : : ; 
n
) near p, with k as above,
precisely ` of the rst k coordinate functions vanish at (p). We usually write
such local coordinates as (x
1
; : : : ; x
k
; y
1
; : : : ; y
n k ). A boundary face of codi-mension ` is the closure of a connected component of @
`X. A boundary face of
codimension 1 is called a boundary hypersurface. A manifold with corners is a
tied manifold with corners such that all boundary hypersurfaces are embedded
submanifolds. This implies the existence of global dening functions H for
*This work is partially supported by NSF grant #DMS-0201092, a fellowship from the
Alfred P. Sloan Foundation and a Clay Research Fellowship.
750 ANDR

AS VASY
each boundary hypersurface H (so that H 2 C
1
(X), H  0, H vanishes
exactly on H and dH 6 = 0 on H); in each local coordinate chart intersecting
H we may take one of the x
j
's (j = 1; : : : ; k ) to be H . While our results are
local, and hence hold for t-manifolds with corners, it is convenient to use the
embeddedness occasionally to avoid overburdening the notation. Moreover, in
a given coordinate system, we often write Hj
for the boundary hypersurface
whose restriction to the given coordinate patch is given by x
j
= 0, so that the
notation Hj
depends on a particular coordinate system having been chosen
(but we usually ignore this point). If X is a manifold with corners, X

denotes
its interior, which is thus a C
1
manifold (without boundary).
Returning to the wave equation, let M be amanifold with corners equipped
with a smooth Riemannian metric g. Let
=

g
be the positive Laplacian of
g, let X = M  Rt
, P = D
2
t

, and consider the Dirichlet boundary condition
for P :
Pu = 0; uj
@X
= 0;
with the boundary condition meaning more precisely that u 2 H
1
0;loc
(X). Here
H
1
0
(X) is the completion of
_
C
1
c
(X) (the vector space of C
1
functions of com-pact support on X, vanishing with all derivatives at @X ) with respect to
kuk
2
H
1
(X)
= kduk
L
2
(X)
+ kuk
L
2
(X)
, L
2
(X) = L
2
(X; dg dt), and H
1
0;loc
(X) is
its localized version; i.e., u 2 H
1
0
(X) if for all  2 C
1
c
(X), u 2 H
1
0
(X). At
the end of the introduction we also consider Neumann boundary conditions.
The statement of the propagation of singularities of solutions has two ad-ditional ingredients: locating singularities of a distribution, as captured by the
wave front set, and describing the curves along which they propagate, namely
the bicharacteristics. Both of these are closely related to an appropropriate
notion of phase space, in which both the wave front set and the bicharacter-istics are located. On manifolds without boundary, this phase space is the
standard cotangent bundle. In the presence of boundaries the phase space is
the b-cotangent bundle,
b
T

X, (`b' stands for boundary), which we now brie
y
describe following [19], which mostly deals with the C
1
boundary case, and
especially [20].
Thus, V
b
(X) is, by denition, the Lie algebra of C
1
vector elds on X
tangent to every boundary face of X. In local coordinates as above, such vector
elds have the form
X
a
j
(x; y)x
j
@
xj
+
X
j
b
j
(x; y)@
yj
with a
j
; bj smooth. Correspondingly, V
b
(X) is the set of all C
1
sections of
a vector bundle
b
TX over X: locally x
j
@
xj
and @
yj
generate V
b
(X) (over
C
1
(X)), and thus (x; y; a; b) are local coordinates on
b
TX