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

Its limit when t → 0is
K. Koufany, G. Zhang / Journal of Functional Analysis 236 (2006) 546–580 573
− 1
(ρ − λ)(ξj−1)R(ζτ(w)).
2
• For the quadratic term ( 1 2ξj − 1 2ζj )(η ¯w − η ′ ¯w ), the corresponding differential operator is
t λ−ρ t − βj −β
j−1 1
2
2 tj
Its limit is
which is
lim
t→0 tλ−ρ t − βj −βj−1 2
= lim
∂
∂tj
t
t→0 λ−ρ t − βj −βj−1 2
− 1
2 tβj
t βj −βj−1 R(ζj ) 2 R(η ¯w) + t βj +βj−1 2 R(η ′ ¯w )t
ρ−λ .
1 ∂ βj −βj−1 t 2 t
2 ∂tj
ρ−λ R(η ¯w) + t βj +βj−1 2 t ρ−λ R(η ′ ¯w )
1 βj − βj−1
(ξj ) + (ρ − λ)(ξj ) t
2 2
βj −βj−1 2 t ρ−λ R(η ¯w)
+ 1
βj + βj−1
(ξj ) + (ρ − λ)(ξj ) t
2 2
βj +βj−1 2 t ρ−λ R(η ′ ¯w )
,
1
1 + (ρ − λ)(ξj )
2
R(η ¯w).
• The induced equation corresponding to the last term of the projection π(Hβ ) is
bR(η ¯w),
(23)
0.
• Now, it is easy to see, using the same computations, that the induced equation of the remaining
terms of π(Hβ ) is zero.
It follows now from (22), (23) and (20), that the boundary value Bλf of f satisfies
where C1 is given by
C1R(ζτ(w))(Bλf)= 0,
C1 = 1
2 (ρ − λ)(ξj
12
+ j + b,
− ξj−1) +
1
2 + j.
Now if C1 = 0, then the induced equation is R(ζτ(w))(Bλf)= 0. On the other hand, if C1 = 0,
we may replace f by tκ(β j −βj−1 2 ) f for sufficiently large κ>0, consider the differential operator
t − 1 2 (βj −βj−1) κ(
t βj −βj−1 2
) β
R π H t −κ(β j −βj−1 2 )
,
and we still prove that R(ζτ(w))(Bλf)= 0, see also [12]. ✷