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 δL=0δL=0, withL[u]:=∫∫(|∂tu|2−W(x;∇xu))dxdt,F↦W(x;F)F↦W(x;F) being a quadratic form over Md×n(R)Md×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 RdRd. 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∫ΩW(∇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 Lopatinskiĭ condition needs only to be satisfied at real   frequency pairs (τ,η)(τ,η) with τ⩾0τ⩾0, instead of pairs with Rτ⩾0. Even stronger is the fact that we need only to examine pairs (τ=0,η)(τ=0,η), and prove that some Hermitian matrix H(η)H(η) is positive definite. Another significant result is that every such well-posed problem admits a pair of surface waves at every frequency η≠0η≠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.