pde9 - 126 BRUCE K DRIVER † 9 Poisson and Laplace’s...

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Unformatted text preview: 126 BRUCE K. DRIVER † 9. Poisson and Laplace’s Equation For the majority of this section we will assume Ω ⊂ 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 ( Ω o ) ∩ C 1 ( Ω ) and R Ω |4 u | dx < ∞ then (9.1) Z Ω 4 u · vdm = − Z Ω ∇ u · ∇ vdm + Z ∂ Ω v ∂u ∂n dσ and if further R Ω {|4 u | + |4 v |} dx < ∞ then (9.2) Z Ω ( 4 uv − 4 v u ) dm = Z ∂ Ω μ v ∂u ∂n − ∂v ∂n u ¶ dσ. Lemma 9.1. Suppose u ∈ C 2 ( Ω o ) ∩ C 1 ( Ω ) , ∆ u = 0 on Ω o and u = 0 on ∂ Ω . Then u ≡ . Similarly if ∆ u = 0 on Ω o and ∂ n u = 0 on ∂ Ω , then u is constant on each connected component of Ω . Proof. Letting v = u in Eq. (9.1) shows in either case that 0 = − Z Ω ∇ u · ∇ udm + Z ∂ Ω u ∂u ∂n dσ = − Z Ω | ∇ u | 2 dm. This then implies ∇ u = 0 on Ω o and hence u is constant on the connected compo- nent of Ω o . If u = 0 on ∂ Ω , 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 ) = ½ 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 | )ˆ x · v where ˆ x = x | x | , ∇ f ( x ) = F ( | x | )ˆ 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 )ˆ x · ∇ g ( r ˆ x ) dx = − Z S n − 1 × [0 , ∞ ) F ( r ) d dr g ( rω ) r n − 1 dr dσ ( ω ) = Z S n − 1 × [0 , ∞ ) d dr ( r n − 1 F ( r )) g ( rω ) dr dσ ( ω ) = Z S n − 1 × [0 , ∞ ) 1 r n − 1 d dr ( r n − 1 F ( r )) g ( rω ) r n − 1 dr dσ ( ω ) = 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 | )ˆ x ] = ∇ F ( | x | ) · ˆ x + F ( | x | ) ∇ · ˆ x = F 00 ( | x | )ˆ x · ˆ x + F ( | x | ) ∇ · x | x | = F 00 ( | x | ) + F ( | x | ) ½ n | x | − x | x | 2 · ˆ x ¾ = F 00 ( | x | ) + ( n − 1) | x | F ( | x | ) . Notation 9.3. For t > , let (9.4) α ( t ) := α n ( t ) := c n ½ 1 t n − 2 if n 6 = 2 ln t if n = 2 , where c n = ½ 1 ( n − 2) σ ( S n − 1 ) if n 6 = 2 − 1 2 π if n = 2 ....
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pde9 - 126 BRUCE K DRIVER † 9 Poisson and Laplace’s...

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