Abstract :
We consider linear hyperbolic boundary-value problems for second order systems, which can be written
in the variational form δL = 0, with
L[u] := |∂tu|2 − W(x;∇xu) dx dt,
F →W(x;F) being a quadratic form overMd×n(R). The domain of L is the homogeneous Sobolev space
H˙ 1(Ω × Rt )n, with Ω either a bounded domain or a half-space of Rd . The boundary condition inherent
to this problem is of Neumann type. Such problems arise for instance in linearized elasticity. When Ω is
a half-space and W depends only on F, we show that the strong well-posedness occurs if, and only if, the
stored energy
Ω W(∇xu) dx
is convex and coercive over H˙ 1(Ω)n. Here, the energy density W does not need to be convex but only
strictly rank-one convex. The “only if” part is the new result. A remarkable fact is that the classical characterization
of well-posedness by the Lopatinski˘ı condition needs only to be satisfied at real frequency pairs
(τ, η) with τ 0, instead of pairs with τ 0. Even stronger is the fact that we need only to examine pairs
(τ = 0,η), and prove that some Hermitian matrix H(η) is positive definite. Another significant result is thatevery such well-posed problem admits a pair of surface waves at every frequency η = 0. These waves often
have finite energy, like the Rayleigh waves in elasticity. When we vary the density W so as to reach nonconvex
stored energies, this pair bifurcates to yield a Hadamard instability. This instability may occur for
some energy densities that are quasi-convex, contrary to the case of the pure Cauchy problem, as shown in
several examples. At the bifurcation, the corresponding stationary boundary-value problem enters the class
of ill-posed problems in the sense of Agmon, Douglis and Nirenberg. For bounded domains and variable
coefficients, we show that the strong well-posedness is equivalent to a Korn-like inequality for the stored
energy.
© 2006 Elsevier Inc. All rights reserved.
