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

442 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
that the linearized problem around a phase boundary has a variational structure too, although of a
slightly different form than the one studied in the present article. Thus the main result of [2], the
existence of finite energy surface waves in this linearized problem is likely to be a consequence
of this structure, in the same spirit as in our Theorem 3.3. We leave this justification for a future
work.
7. General domain and variable coefficient
We turn towards the realistic case where Ω is a smooth open domain in Rd and the quadratic
energy density W(x;∇xu) may depend smoothly upon the space variable. For the sake of simplicity,
we limit ourselves to bounded domains. The smoothness required for the boundary and
the coefficients is C2 . The Lagrangian
L[u]:= |∂tu| 2 − W(x;∇xu) dxdt
Ω
defines an initial boundary-value problem in Ω. From this IBVP, we define a family of IBVPs
with constant coefficients in half-spaces, parametrized by the elements of the boundary ∂Ω.To
each point x0 ∈ ∂Ω, we associate the Lagrangian
L[u]:= |∂tu| 2 − W(x0;∇xu) dxdt,
ω(x0)
where ω(x0) is the half-space with the same outer normal ν(x0) at x0 as Ω:
ω(x0) ={x: (x − x0) · ν(x0)