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lecture7_001

lecture7_001 - Lecture Seven Consequences of Cacciopolli 1...

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1 Lecture Seven: Consequences of Cacciopolli Consequences of Cacciopolli In this lecture we continue to generalise some of our results about the Laplacian to uni- formly elliptic second order operators. We start by stating two results without proof. Proposition 1.1 Let B 2 r be a ball in R n , and let L be a uniformly elliptic second order operator with Lf = · A f, and λ | v | 2 A v Λ v 2 . There are positive constants c , and d depending only on the v · dimension and the ratio | λ | such that Λ 2 2 u (1 + c ) u (1) B 2 r B r and 2 2 (1 + d ) |� u | |� u | (2) B 2 r B r for all u with Lu = 0 on B 2 r . This is very similar to a result for harmonic functions from Lecture 5, but we will not give the proof here. Instead we will work on some of the consequences. it is clear from (1) that if u is L harmonic (ie Lu = 0) on B 2 m s then 2 2 u (1 + c ) m u . (3) B 2 m s B s log(1+ c ) m log(1+ c ) = 2 m log 2 We can rewrite (1 + c ) m = e . Define α = log(1+ c ) > 0, and let t = 2 m s .

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lecture7_001 - Lecture Seven Consequences of Cacciopolli 1...

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