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

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372 H.-Q. Li / Journal of Functional Analysis 236 (2006) 369–394 très faibles lorsque t →+∞. Dans certains cas, la méthode de cet article nous permet d’obtenir des estimations optimales de type (1.2) lorsque t →+∞. Pour mieux comprendre les avantages de notre méthode (qui consiste à utiliser directement des estimations du noyau de la chaleur et de son gradient, plutôt que la formule de Bochner- Lichnerowicz, comme [4]), considérons une variété riemannienne compacte, sans bord et de dimension n, N. Notons pt le noyau de la chaleur sur N, λ1 > 0 la première valeur propre non nulle du Laplacien et |N| son volume. Alors pour t ≫ 1, on a d’une part : ∇e t f(g) = ∇ N N pt(g, g ′ ) f(g ′ ) − 1 |N| N N f(g∗)dμ(g∗) dμ(g ′ ) ∇pt(g, g ′ ) · f(g′ ) − 1 f(g∗)dμ(g∗) |N| dμ(g′ ) C1e −λ1t f(g′ ) − 1 f(g∗)dμ(g∗) |N| dμ(g′ ) N N par le fait que ∇pt(g, g ′ ) C1e−λ1t pour tout t ≫ 1etg,g ′ ∈ N C2e −λ1t diam(N) |∇f |(g ′ )dμ(g ′ ) N par l’inégalité de Poincaré Ce −λ1t diam(N)|N| pt(g, g ′ )|∇f |(g ′ )dμ(g ′ ) N par le fait que pt(g, g ′ ) 2 −1 |N| −1 pour tout t ≫ 1etg,g ′ ∈ N Ce −λ1t diam(N)|N| e t |∇f | w (g) 1/w , pour tout g ∈ N, f ∈ C ∞ (N) et 1 we ∇υ1Lw −1/w 2 , |N| puisque pt(g, g ′ ) 2|N| −1 pour tout t ≫ 1etg,g ′ ∈ N. Autrement dit, dans le cadre des variétés riemanniennes compactes, sans bord, on a sup g∈N,f∈C ∞ (N) |∇e t f(g)| [e t (|∇f | w )(g)] 1/w ∼ e−λ1t .
H.-Q. Li / Journal of Functional Analysis 236 (2006) 369–394 373 Et on remarque que lorsque t →+∞, le facteur e−λ1t est décroissant beaucoup plus vite que e− max(0,ko)t avec ko = sup{k ∈ R; Ric(N) k}, en rappelant que le théorème de Lichnerowicz nous dit que λ1 n n−1ko pour ko > 0. 2. Dans [21,23], les estimations (1.4) avec 1
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3.4. The zeroes of ( √ z, η) D.
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4.1. Persistence of surface waves D
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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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Journal of Functional Analysis 236
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we now have (cf. (7.8)) that H. Sch
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2.2. Definition J. Van Schaftingen
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Since ϕ = ϕ0 + ϕ1 ∧ ωN, one h
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3. The Poisson transform K. Koufany
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Its limit when t → 0is K. Koufany
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and K. Koufany, G. Zhang / Journal
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Using it, we obtain as in (3.2) tha
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The minimum is realized for S. Albe
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hence S. Albeverio, A. Kosyak / Jou
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φpq(t; tqq) = Since = S. Albeveri
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hence C = C3 = S. Albeverio, A. Kos
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lim n→∞ Ω lim sup n→∞
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