Download Analysis of harmonic maps and their heat flows by Lin F., Wang C. PDF

By Lin F., Wang C.

ISBN-10: 9812779523

ISBN-13: 9789812779526

This publication presents a wide but accomplished creation to the research of harmonic maps and their warmth flows. the 1st a part of the publication includes many vital theorems at the regularity of minimizing harmonic maps by means of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in greater dimensions via Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by means of Helein, in addition to at the constitution of a novel set of minimizing harmonic maps and desk bound harmonic maps via Simon and Lin.The moment a part of the booklet features a systematic insurance of warmth move of harmonic maps that incorporates Eells-Sampson's theorem on international gentle recommendations, Struwe's nearly average strategies in measurement , Sacks-Uhlenbeck's blow-up research in measurement , Chen-Struwe's life theorem on in part gentle recommendations, and blow-up research in better dimensions by means of Lin and Wang. The booklet can be utilized as a textbook for the subject process complex graduate scholars and for researchers who're drawn to geometric partial differential equations and geometric research.

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The argument was implicitly contained in the previous section. For the convenience of readers, we sketch it. First note that if u ∈ Q(φ, ˆ, λ) then v(x) = u(8λ 3 x) ∈ Q φ, ˆ, 21 . Hence if we verify that v ∈ Q φ, 1, µ2 , then u ∈ Q(φ, 1, µλ). Thus we may assume that λ = 12 . Suppose the conclusion were false. 20) and Θ n−2 (ui , 0) = 2 B1 |x|2−n ∂ui ∂r i ↓ 0. 6. 74) since ES n−1 (·) is uniformly C 2 in v ∈ C 2 (S n−1 , N ) : v − φ C 2 (S n−1 ) ≤ 0 for some sufficiently small 0 > 0. 73) 1 ui (τ ) − ui (1) L2 (S n−1 ) ≤ ∂ui ∂r τ 1 ≤ τ dr L2 (S n−1 ) 1 2 dr r r 2−n B1 1 ≤ |log τ | 2 ui − φ ∂ui ∂r 1 2 2 1 1,2−3 ≤ |log τ | 2 i.

5 For n ≥ 3, if u ∈ W 1,2 (Ω, N ) is a minimizing harmonic map, then for any x ∈ Ω and 0 < r ≤ R < dist(x, ∂Ω), R2−n BR (x) ≥ |∇u|2 − r 2−n BR (x)\Br (x) |y − x|2−n Br (x) |∇u|2 ∂u ∂|y − x| 2 . 20) rx Proof. For simplicity, assume x = 0. For r > 0, let u r (x) = u( |x| ) for x ∈ Br . Then, by the minimality, we have Br |∇u|2 ≤ Br |∇ur |2 r = r2 tn−3 S n−1 0 = = r n−2 r n−2 ∂Br |∇S n−1 u|2 (rθ) dH n−1 (θ) |∇T u|2 dH n−1 |∇u|2 − | ∂Br ∂u 2 | ∂r dH n−1 , where ∇S n−1 and ∇T denote the gradient on S n−1 and ∂Br respectively.

We now recall that if L is a j-dimensional subspace of R n and for each δ ∈ (0, 18 ) we can find β = β(δ) with limδ→0 β(δ) = 0 and σ = σ(δ) ∈ (0, 1) such that for each R > 0 a 2δR-neighborhood of L ∩ BR (0) can be covered by balls BδR (yk ) with centers yk ∈ L ∩ BR (0), k = 1, · · · , Q such that Q(δR)j+β(δ) < 21 Rj+β(δ) . Now the lemma follows by using successively finer covers of A by balls. For simplicity assume A is bounded, we first take an initial cover of A by balls B ρ0 (yk ) 2 with A ∩ B ρ0 (yk ) = ∅, k = 1, · · · , Q, and let T0 = 21 ( ρ20 )j+β(δ) .

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