Appl Math Optim 44:33–48 (2001)
2001 Springer-Verlag New York Inc.
On a Stochastic Plate Equation
J. U. Kim
Department of Mathematics, Virginia Tech,
Blacksburg, VA 24061-0123, USA
Abstract. In this paper we discuss an initial–boundary valueproblem for an elastic
plate driven by a space-time white noise. The existence and uniqueness of a weak
solution is established. We use a specialized PDE method based upon the results for
the deterministic equation.
Key Words. Plate equation, Weak solution, White noise, Stochastic integral.
AMS Classiﬁcation. 35Q20, 60H15, 73K12.
In this paper we discuss an initial–boundary value problem for an elastic plate equation
with a space-time white noise forcing. When the deﬂection of the plate from equilibrium
is denoted by u = u(t, x), the model equation is given by
+ (t )u = σ(t, x, u)
W , for (t, x) ∈ (0, T ) × D, (0.1)
where t is the time variable, x = (x
) is the space variable, and m(x) is the mass per
unit area of the plate. D is a bounded domain in R
with smooth boundary ∂ D, which
represents the plate. (t) is a differential operator deﬁned by
(t) = (a(t, x)) +
where is the Laplacian in x, and the coefﬁcients a(t, x) and b
(t, x) are given func-
tions. a(t, x) stands for the ﬂexural rigidity of the plate deﬁned by
a(t, x) =
12(1 − ν