Second order initial boundary-value problems of variational type

We consider linear hyperbolic boundary-value problems for second order systems, which can be written in the variational form delta L = 0, with L[u] := integral integral (vertical bar(t)u vertical bar(2) - W (x; del(x)u))dxdt, F -> W(x; F) being a quadratic form over M-dxn (R). The domain of L is the homogeneous Sobolev space H-1 (Omega x R-t)(n), with Omega either a bounded domain or a half-space of R-d. The boundary condition inherent to this problem is of Neumann type. Such problems arise for instance in linearized elasticity. When Omega 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 integral W-Omega(del(x)u)dx is convex and coercive over H-1(Omega)(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 Lopatinskii condition needs only to be satisfied at real frequency pairs (tau,eta) with tau >= 0, instead of pairs with R-tau >= 0. Even stronger is the fact that we need only to examine pairs (tau = 0,eta), and prove that some Hermitian matrix H(eta) is positive definite. Another significant result is that every such well-posed problem admits a pair of surface waves at every frequency eta not equal 0. These waves often have finite energy, like the Rayleigh waves in elasticity. When we vary the density W so as to reach non-convex 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. (c) 2006 Elsevier Inc. All rights reserved.