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
where the solution to the
wave equation (with zero initial position) is zero for all time. Stationary sets,
depend on the initial velocity,
We also consider sets of points,
where the solution vanishes starting from a certain moment in time
depends on the point
For the wave equation in even dimensional
and any function
T : IRn
we prove that
when the initial velocity
has compact support, and we prove some support
in odd dimensions.
We also study stationary sets in the case when the outer boundary of
is convex and prove facts that support a conjecture raised earlier by
the authors which states that stationary hypersurfaces are cones.
Let us consider the following initial value problem for the wave equation in JR.n:
u(x, t), x
is in the space of continuous functions of compact support,
LetT: JR.n----+ [0, oo), and define the T-stationary set
is the set of all points in
JR.n where the
t) to (1.1) vanishes for all timet
is identically zero
0 but with initial velocity given by
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
is the intersec-
tion of nodal sets, (e.g.,
Lin and Pinkus originally considered this problem in
relation to approximation theory
and there are many results on the structure
does not grow too large at infinity, Agranovsky, Berenstein, and Kuch-
contains no closed bounded surface provided
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