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

D. Serre / Journal of Functional Analysis 236 (2006) 409–446 443
This result follows immediately from the Hille–Yosida theorem and the following estimate.
Theorem 7.2. Under the assumptions of Theorem 7.1, there exists two positive constants ɛ and C,
such that
Proof. If x1 ∈ Ω, let us define
W[u] ɛ∇xu 2
L2 − Cu2
(ω) L2 (ω) .
Y[x1; u]:=
R d
W(x1;∇xu) dx.
By rank-one convexity, there exists a positive number α(x1) such that Y[x1; u] α(x1)∇xu 2 ,
where ·stands for the L 2 (R d )-norm. Since W varies smoothly with x, and since Ω is compact,
we have
inf
x1∈Ω α(x1)>0.
Likewise, Theorem 3.5 tells that if x0 is a boundary point, then W[x0; u] dominates
β(x0) ∇xu 2 dx
ω(x0)
for some positive number β(x0). Again, continuity and compactness of the boundary imply
inf
x0∈∂Ω β(x0)>0.
In short, there exists a number γ>0 such that for every interior point x1 or boundary point x0,
there holds true
Y[x1; u] γ ∇xu 2 , W[x0; u] γ ∇xu 2 ,
where we employ the L2-norm, either of Rd or the half-space ω(x0), according to the context.
Let the ball B = B(x1; r) be contained in Ω. Ifu∈ H 1 (Ω) has support contained in B,
extension by zero yields a ũ ∈ H˙ 1 (Rd ). Then
W[u]=Y[x1;ũ]+ W(x;∇xu) − W(x1;∇xu) dx.
Because of smoothness, we derive
B
W[u] γ ∇xu 2 − O(r)∇xu 2 .
Therefore, choosing r small enough, we are certain that
W[u] γ 2
∇xu
2