By convention whenever n > 1 and x, y e R", we write |x| = Jx > X;Yi- +...+ x; and (x, y) = i=1 1. Consider Q = {x € R? : [x| < 1} and Q* = {x e R² : [x| > 1}. For any x e R² \ {0}, set F(x) = x* = . For any function u : Q → R, we define a function u* : Q* → R setting u*(x) = u (x*), Vx € Q°. a) Show that F;(x) = and F2(x) = are harmonic in R² \ {0} with |VF;(x)| = |VF2(x)| = 1 and (VF¡(x), VF2(x)) = 0.
By convention whenever n > 1 and x, y e R", we write |x| = Jx > X;Yi- +...+ x; and (x, y) = i=1 1. Consider Q = {x € R? : [x| < 1} and Q* = {x e R² : [x| > 1}. For any x e R² \ {0}, set F(x) = x* = . For any function u : Q → R, we define a function u* : Q* → R setting u*(x) = u (x*), Vx € Q°. a) Show that F;(x) = and F2(x) = are harmonic in R² \ {0} with |VF;(x)| = |VF2(x)| = 1 and (VF¡(x), VF2(x)) = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
part a), please
![By convention whenever n > 1 and x, y e R", we write
|x| = Jx + ... + x? and (x, y) = > x;Yi-
i=1
1. Consider Q = {x € R? : ]x| < 1} and Qª = {x € R? : |x| > 1}. For any x e R² \ {0}, set
= . For any function u : 2 → R, we define a function u* : Q* → R setting
%3D
|x? *
u*(x) = u (x*), Vx e N*.
a) Show that F1(x) = and F2(x) = are harmonic in R² \ {0} with
1
and (VF¡(x), VF2(x)) = 0.
|x?
|VF;(x)| = |VF2(x)| =
b) For any u e C²(N), show that u is sub-harmonic in 2 \ {0} if and only if u* e C²(N*)
is sub-harmonic in Q*.
Hint: Show first the identity
A(u o F) = VF, VF,)ð; ju o F + A(F;)a;u o F.
i.j
и(х)
c) Assume that u e C²(Q*) n C°Q*) is harmonic such that lim>0
U. = lim-r0 u(x) exists with
= 0. Show that
27
1
U0o =
2л
u(cos(0), sin(0))d0.
Hint: You may use the removable singularity theorem.
d) Show that there is no harmonic function u e C²(N°) N C°(N*) with lim¬00 U(x) = 0
and u(w) = 1 for all w with |w| = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d02b6-78f1-4c0a-a713-3b04a8400050%2Fd288a4c1-635e-445b-b09d-86bed4027473%2F1y5gd5_processed.png&w=3840&q=75)
Transcribed Image Text:By convention whenever n > 1 and x, y e R", we write
|x| = Jx + ... + x? and (x, y) = > x;Yi-
i=1
1. Consider Q = {x € R? : ]x| < 1} and Qª = {x € R? : |x| > 1}. For any x e R² \ {0}, set
= . For any function u : 2 → R, we define a function u* : Q* → R setting
%3D
|x? *
u*(x) = u (x*), Vx e N*.
a) Show that F1(x) = and F2(x) = are harmonic in R² \ {0} with
1
and (VF¡(x), VF2(x)) = 0.
|x?
|VF;(x)| = |VF2(x)| =
b) For any u e C²(N), show that u is sub-harmonic in 2 \ {0} if and only if u* e C²(N*)
is sub-harmonic in Q*.
Hint: Show first the identity
A(u o F) = VF, VF,)ð; ju o F + A(F;)a;u o F.
i.j
и(х)
c) Assume that u e C²(Q*) n C°Q*) is harmonic such that lim>0
U. = lim-r0 u(x) exists with
= 0. Show that
27
1
U0o =
2л
u(cos(0), sin(0))d0.
Hint: You may use the removable singularity theorem.
d) Show that there is no harmonic function u e C²(N°) N C°(N*) with lim¬00 U(x) = 0
and u(w) = 1 for all w with |w| = 1.
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