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

444 D. Serre / Journal of Functional Analysis 236 (2006) 409–446
holds true. We point out that the same positive r may be chosen for every point x1. To treat the
case of a boundary point x0, we employ the same argument, but we need another technical tool.
If r is small enough, then B ∩ Ω is diffeomorphic to a half-ball
B+ := x: (x − x0) · ν(x0)0 as above. By compactness we may cover Ω by a finite collection of balls Bj :=
B(x j ; r) where either Bj ⊂ Ω or x j ∈ ∂Ω (here it is useful to take r less than the minimum of
the curvature of the boundary). Let (ρ1,...,ρN) be a partition of unity over Ω adapted to the
B ′ j s, with ρj 0. If u ∈ H 1 (Ω), we define
Using the polar form Ψ of W ,wehave
uj := ρj u, ujk := √ ρj ρk u.