By convention whenever n ≥ 1 and x, y € R", we write |x| = + … + x², and (x, y) = ΣXiyi. i=1 n 1. Consider = {x € R² : [x] < 1} and №* = {x € R² : |x| > 1}. For any x € R² \ {0}, set F(x) = x² =. For any function u : Q→ R, we define a function u* : Q* → R setting u*(x) = u(x*), \x € *. a) Show that F₁(x) = 2 and F₂(x) = |VF₁(x)| = |VF₂(x)| = = are harmonic in R² \ {0} with 1 |x|² and (VF₁(x), VF₂(x)) = 0. b) For any u € C²(Q), show that u is sub-harmonic in \ {0} if and only if u* € C²(N*) is sub-harmonic in Q*. Hint: Show first the identity ▲(u o F) = Σ(VF ¡, VF j)ði¡ ju ○ F + Σ A(F ;)ð ¡u ○ F. i.j j
By convention whenever n ≥ 1 and x, y € R", we write |x| = + … + x², and (x, y) = ΣXiyi. i=1 n 1. Consider = {x € R² : [x] < 1} and №* = {x € R² : |x| > 1}. For any x € R² \ {0}, set F(x) = x² =. For any function u : Q→ R, we define a function u* : Q* → R setting u*(x) = u(x*), \x € *. a) Show that F₁(x) = 2 and F₂(x) = |VF₁(x)| = |VF₂(x)| = = are harmonic in R² \ {0} with 1 |x|² and (VF₁(x), VF₂(x)) = 0. b) For any u € C²(Q), show that u is sub-harmonic in \ {0} if and only if u* € C²(N*) is sub-harmonic in Q*. Hint: Show first the identity ▲(u o F) = Σ(VF ¡, VF j)ði¡ ju ○ F + Σ A(F ;)ð ¡u ○ F. i.j j
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
![By convention whenever n > 1 and x, y e R", we write
|x| = Vx +...+ x, and (x, y)
i=1
1. Consider Q = {x € R² : ]x[ < 1} and Q* = {x e R² : |x| > 1}. For any x e R² \ {0}, set
F(x) = x* = . For any function u : 2 → R, we define a function u* : Q* →
%3D
%3D
R setting
u*(x) = u (x*), Vx e N*.
a) Show that F;(x) = and F2(x) = are harmonic in R² \ {0} with
%3D
%3D
|VF;(x)| = |VF2(x)| =
1
and (VF1(x), VF2(x)) = 0.
b) For any u € C²(N), show that u is sub-harmonic in 2 \ {0} if and only if u e C²(Q*)
is sub-harmonic in Q*.
Hint: Show first the identity
A(u o F) = VF;, VF;)ð;ju o F + a(F )ð ju o F.
ij](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad4d02b6-78f1-4c0a-a713-3b04a8400050%2F2f0e90de-f868-4e29-a8f5-112af6cb15d9%2F38oguus_processed.png&w=3840&q=75)
Transcribed Image Text:By convention whenever n > 1 and x, y e R", we write
|x| = Vx +...+ x, and (x, y)
i=1
1. Consider Q = {x € R² : ]x[ < 1} and Q* = {x e R² : |x| > 1}. For any x e R² \ {0}, set
F(x) = x* = . For any function u : 2 → R, we define a function u* : Q* →
%3D
%3D
R setting
u*(x) = u (x*), Vx e N*.
a) Show that F;(x) = and F2(x) = are harmonic in R² \ {0} with
%3D
%3D
|VF;(x)| = |VF2(x)| =
1
and (VF1(x), VF2(x)) = 0.
b) For any u € C²(N), show that u is sub-harmonic in 2 \ {0} if and only if u e C²(Q*)
is sub-harmonic in Q*.
Hint: Show first the identity
A(u o F) = VF;, VF;)ð;ju o F + a(F )ð ju o F.
ij
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