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

414 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
2. Convex stored energies
The role of convex stored energies is fundamental in the study of the homogeneous IBVP. As
mentioned in the introduction, there is a balance law for the total energy:
∂t
1
2 |∂tu| 2
+ W(∇xu) −
d
n
∂α
α=1 j=1
∂W
∂tuj
∂Fαj
(∇xu) = (∂tu) · f. (6)
When W is coercive on ˙
H 1 (Ω) n , the well-posedness is ensured by the Hille–Yosida theorem and
Duhamel’s formula, as long as g ≡ 0. Clearly, coerciveness implies strict convexity, although the
converse is not true.
The addition of a null-form to W modifies the stored energy W in general, because the integral
of the minor ∂duj ∂αuk − ∂duk∂αuj over Ω equals a boundary integral. Such an addition,
although leaving the differential operator P unchanged, does modify the boundary operator B.
For this reason, when dealing with a specific IBVP, we are only free to add a tangential null-form
(TNF) to W . This terminology designates the linear combinations of minors Fαj Fβk − FαkFβj
when 1 α