Also note that u2 u u2 2u2 2u2 by cauchyschwarz

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Unformatted text preview: tart of this lecture. Recall from the proof of the Cacciopolli inequality last time that if u is L harmonic on B2r then � �2 � � Λ 2 2 φ2 |�u|2 u |�φ| ≥ 4 λ B2r B2r for all φ ≥ 0 with φ = 0 on the boundary. Also note that |�(φu)|2 = |u�φ + φ�u|2 ≤ 2|u�φ2 | + 2|φ�u|2 by Cauchy­Schwarz. Putting these together gives � B2r 2 (10) |�(φu)| 2|u�φ2 | + 2|φ�u|2 B2r �� � �� Λ2 ≤24 +1 u2 |�φ|2 . λ B2r ≤ � (11) (12) Notice that φu is zero on the boundary of B2r , so the Dirichlet­Poincare applies,...
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This note was uploaded on 05/01/2012 for the course MATH 2373 taught by Professor Miracle during the Spring '08 term at Minnesota.

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