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

416 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
Lemma 2.1. The stored energy W is convex on H 1 (Ω) n if, and only if, the following onedimensional
quadratic functional Iη is non-negative over H 1 (0, +∞) n , for every vector η ∈
R d−1 :
Notice that when v = Fyu, then
Iη[v]:=
+∞
0
W(iη⊗ v,v ′ )dxd.
Fy(∇yu) = iη ⊗ v, Fy(∂du) = v ′ .
We are thus led to the study of the convexity over H 1 (0, +∞) n of functionals of the form
I[v]:= 1
2
+∞
0
w(v,v ′ )dxd, (8)
where w : C n × C n → R is a sesquilinear form. In addition, the densities w satisfy
∃ɛ >0 s.t. w(v,iξv) ɛ |η| 2 + ξ 2 |v| 2 , (9)
because of uniform rank-one convexity. Notice that if W1 and W2 differ only by a TNF, then
w1 = w2.
Let us decompose a general density w as
w(v,v ′ ) =〈Λv ′ ,v ′ 〉+2ℜ〈Av ′ ,v〉+〈Σv,v〉, (10)
where 〈·,·〉 is the Hermitian product in C n and Λ and Σ are Hermitian positive definite, because
of (9). We point out that in our variational context, Λ, Σ = Ση and iA = iAη have real entries.
Here, we give a sufficient condition for the convexity of W, which will be shown necessary
in Proposition 3.2.
Theorem 2.2. Let the functional I be defined by (8), (10). Assume that there exists a non-negative
Hermitian n × n matrix K, with the property that the (2n) × (2n) Hermitian matrix
Σ A+ K
S :=
A∗
(11)
+ K Λ
be non-negative. Then I is convex over H 1 (0, +∞) n .
If S is positive definite, then I is coercive.
Proof. Let wS be the sesquilinear form associated to S. Then
I[v]= 1
2
+∞
wS(v, v ′ ) − 2ℜ〈Kv ′ ,v〉 dxd.
0