pde9 - 126 BRUCE K. DRIVER 9. Poisson and Laplaces Equation...

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Unformatted text preview: 126 BRUCE K. DRIVER 9. Poisson and Laplaces Equation For the majority of this section we will assume R n is a compact manifold with C 2 boundary. Let us record a few consequences of the divergence theorem in Proposition 8.28 in this context. If u, v C 2 ( o ) C 1 ( ) and R |4 u | dx < then (9.1) Z 4 u vdm = Z u vdm + Z v u n d and if further R {|4 u | + |4 v |} dx < then (9.2) Z ( 4 uv 4 v u ) dm = Z v u n v n u d. Lemma 9.1. Suppose u C 2 ( o ) C 1 ( ) , u = 0 on o and u = 0 on . Then u . Similarly if u = 0 on o and n u = 0 on , then u is constant on each connected component of . Proof. Letting v = u in Eq. (9.1) shows in either case that 0 = Z u udm + Z u u n d = Z | u | 2 dm. This then implies u = 0 on o and hence u is constant on the connected compo- nent of o . If u = 0 on , these constants must all be zero. Proposition 9.2 (Laplacian on radial functions) . Suppose f ( x ) = F ( | x | ) , then (9.3) 4 f ( x ) = 1 r n 1 d dr ( r n 1 F ( r )) r = | x | = F 00 ( | x | ) + ( n 1) | x | F ( | x | ) . In particular F ( | x | ) = 0 implies d dr ( r n 1 F ( r )) = 0 and hence F ( r ) = Ar 1 n . That is to say F ( r ) = Ar 2 n + B if n 6 = 2 A ln r + B if n = 2 . PDE LECTURE NOTES, MATH 237A-B 127 Proof. Since ( v f )( x ) = F ( | x | ) v | x | = F ( | x | ) x v where x = x | x | , f ( x ) = F ( | x | ) x . Hence for g C 1 c ( R n ) , Z R n 4 f ( x ) g ( x ) dx = Z R n f ( x ) g ( x ) dx = Z R n F ( r ) x g ( r x ) dx = Z S n 1 [0 , ) F ( r ) d dr g ( r ) r n 1 dr d ( ) = Z S n 1 [0 , ) d dr ( r n 1 F ( r )) g ( r ) dr d ( ) = Z S n 1 [0 , ) 1 r n 1 d dr ( r n 1 F ( r )) g ( r ) r n 1 dr d ( ) = Z R n 1 r n 1 d dr ( r n 1 F ( r )) r = | x | g ( x ) dx. Since this is valid for all g C 1 c ( R n ) , Eq. (9.3) is valid. Alternatively, we may simply compute directly as follows: 4 f ( x ) = [ F ( | x | ) x ] = F ( | x | ) x + F ( | x | ) x = F 00 ( | x | ) x x + F ( | x | ) x | x | = F 00 ( | x | ) + F ( | x | ) n | x | x | x | 2 x = F 00 ( | x | ) + ( n 1) | x | F ( | x | ) . Notation 9.3. For t > , let (9.4) ( t ) := n ( t ) := c n 1 t n 2 if n 6 = 2 ln t if n = 2 , where c n = 1 ( n 2) ( S n 1 ) if n 6 = 2 1 2 if n = 2 ....
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pde9 - 126 BRUCE K. DRIVER 9. Poisson and Laplaces Equation...

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