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