Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

Journal of Functional Analysis 236 (2006) 409–446
www.elsevier.com/locate/jfa
Second order initial boundary-value problems of
variational type
Denis Serre
École Normale Supérieure de Lyon, France 1
Received 29 August 2005; accepted 26 February 2006
Available online 18 April 2006
Communicated by H. Brezis
Abstract
We consider linear hyperbolic boundary-value problems for second order systems, which can be written
in the variational form δL = 0, with
|∂t
L[u]:= u| 2 − W(x;∇xu) dx dt,
F ↦→ W(x; F)being a quadratic form over 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 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
W(∇xu) dx
Ω
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) with τ 0, instead of pairs with ℜτ 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 that
E-mail address: denis.serre@umpa.ens-lyon.fr.
1 UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, F-69364 Lyon, cedex 07, France. The research of the
author was partially supported by the European IHP project “HYKE”, contract # HPRN-CT-2002-00282.
0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jfa.2006.02.020