Contemporary Mathematics
Volume 405, 2006
Remarks on stationary sets for the wave equation
Mark L. Agranovsky and Eric Todd Quinto
ABSTRACT. Stationary sets are sets of points in
IRn
where the solution to the
wave equation (with zero initial position) is zero for all time. Stationary sets,
S[f],
depend on the initial velocity,
f(x).
We also consider sets of points,
Sr[J],
where the solution vanishes starting from a certain moment in time
T(x)
that
depends on the point
x
E
IRn.
For the wave equation in even dimensional
space,
JR2n,
and any function
T : IRn
-+
[0,
CXJ),
we prove that
Sr [!]
=
S[f]
when the initial velocity
f
has compact support, and we prove some support
restrictions on
f
in odd dimensions.
We also study stationary sets in the case when the outer boundary of
supp
f
is convex and prove facts that support a conjecture raised earlier by
the authors which states that stationary hypersurfaces are cones.
1. Introduction
Let us consider the following initial value problem for the wave equation in JR.n:
f:l.u
=
Utt
u
=
u(x, t), x
E
lRn,
t
0,
(1.1)
u(x,O)
=
0,
x
E
lRn,
Ut(X,
0)
=
j(x),
X
E
JR.n,
where
f
is in the space of continuous functions of compact support,
f
E
Cc(lRn).
LetT: JR.n----+ [0, oo), and define the T-stationary set
Sr[f]
as
(1.2)
Sr[f]
=
{x
E
JR.n:
u(x, t)
=
0, V
t T(x)}.
We define
S[f]
=
S0 [f].
Thus,
Sr[f]
is the set of all points in
x
E
JR.n where the
solution
u(x,
t) to (1.1) vanishes for all timet
T(x)
assuming
u
is identically zero
at
t
=
0 but with initial velocity given by
f(x).
This problem has a rich history, in part due to its relation to nodal sets for
eigenfunctions for the Laplacian [8, 9, 11, 12]. One can show
S[f]
is the intersec-
tion of nodal sets, (e.g.,
[5]).
Lin and Pinkus originally considered this problem in
relation to approximation theory
[16],
and there are many results on the structure
of
S[f]. Iff
does not grow too large at infinity, Agranovsky, Berenstein, and Kuch-
ment
[2]
showed that
S[f]
contains no closed bounded surface provided
f
decays
2000
Mathematics Subject Classification.
Primary: 35105, 44A12 Secondary: 35B05, 35S30.
The first author was supported by the Israel Science Foundation (grant No.279/02-l).
The second author was partially supported by Tufts University FRAC and the National
Science Foundation under grants DMS-0200788 and DMS-0456858.
@2006 Mark L. Agranovsky and Eric Todd Quinto
http://dx.doi.org/10.1090/conm/405/07609
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