A Question About Laplace Equation with U={|x|>1}
$begingroup$
I'm trying to prove the question below. And I'm thinking about using Maximum Principle to prove it. However, U here is not a bounded region. Additionally, for the energy method, I cannot get an idea to apply since integration by parts doesn't work here. Also, there is no condition that the limit of u goes to zero as x goes to infinity...
Could anyone give me some ideas about this question? Thanks a lot!
Let $U = {x ∈ R^n: |x| > 1}$, Suppose $u ∈ C^2(U) ∩ C(overline{U})$ is a bounded solution of the following Dirichlet problem: $∆u = 0$ in $U$ and $u = ϕ$ on $Γ = {x ∈ R^n: |x| = 1}$, with $ϕ ∈ C(Γ)$
a) If n = 2, show that there exists at most one solution of the above problem
b) If n=3, show that it is possible to have more than one bounded solutions of the above problem. What additional condition should you impose so that the solution is unique?
pde
asked Dec 23 '18 at 1:56
WangWang
111
$endgroup$
add a comment |
$begingroup$
I'm trying to prove the question below. And I'm thinking about using Maximum Principle to prove it. However, U here is not a bounded region. Additionally, for the energy method, I cannot get an idea to apply since integration by parts doesn't work here. Also, there is no condition that the limit of u goes to zero as x goes to infinity...
Could anyone give me some ideas about this question? Thanks a lot!
Let $U = {x ∈ R^n: |x| > 1}$, Suppose $u ∈ C^2(U) ∩ C(overline{U})$ is a bounded solution of the following Dirichlet problem: $∆u = 0$ in $U$ and $u = ϕ$ on $Γ = {x ∈ R^n: |x| = 1}$, with $ϕ ∈ C(Γ)$
a) If n = 2, show that there exists at most one solution of the above problem
b) If n=3, show that it is possible to have more than one bounded solutions of the above problem. What additional condition should you impose so that the solution is unique?
pde
asked Dec 23 '18 at 1:56
WangWang
111
$endgroup$
$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49
add a comment |
$begingroup$
I'm trying to prove the question below. And I'm thinking about using Maximum Principle to prove it. However, U here is not a bounded region. Additionally, for the energy method, I cannot get an idea to apply since integration by parts doesn't work here. Also, there is no condition that the limit of u goes to zero as x goes to infinity...
Could anyone give me some ideas about this question? Thanks a lot!
Let $U = {x ∈ R^n: |x| > 1}$, Suppose $u ∈ C^2(U) ∩ C(overline{U})$ is a bounded solution of the following Dirichlet problem: $∆u = 0$ in $U$ and $u = ϕ$ on $Γ = {x ∈ R^n: |x| = 1}$, with $ϕ ∈ C(Γ)$
a) If n = 2, show that there exists at most one solution of the above problem
b) If n=3, show that it is possible to have more than one bounded solutions of the above problem. What additional condition should you impose so that the solution is unique?
pde
asked Dec 23 '18 at 1:56
WangWang
111
$endgroup$
I'm trying to prove the question below. And I'm thinking about using Maximum Principle to prove it. However, U here is not a bounded region. Additionally, for the energy method, I cannot get an idea to apply since integration by parts doesn't work here. Also, there is no condition that the limit of u goes to zero as x goes to infinity...
Could anyone give me some ideas about this question? Thanks a lot!
Let $U = {x ∈ R^n: |x| > 1}$, Suppose $u ∈ C^2(U) ∩ C(overline{U})$ is a bounded solution of the following Dirichlet problem: $∆u = 0$ in $U$ and $u = ϕ$ on $Γ = {x ∈ R^n: |x| = 1}$, with $ϕ ∈ C(Γ)$
a) If n = 2, show that there exists at most one solution of the above problem
b) If n=3, show that it is possible to have more than one bounded solutions of the above problem. What additional condition should you impose so that the solution is unique?
pde
pde
asked Dec 23 '18 at 1:56
WangWang
111
asked Dec 23 '18 at 1:56
WangWang
111
asked Dec 23 '18 at 1:56
WangWang
111
asked Dec 23 '18 at 1:56
WangWang
111
asked Dec 23 '18 at 1:56
WangWang
111
111
$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49
add a comment |
$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49
$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49
$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49
add a comment |
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$begingroup$
Also, I'm thinking about using Kelvin transorm... But for n=2, how to prove the transformed u* is harmonic by using the boundedness of u. Also, I still don't have an idea about n=3 counterexample.
$endgroup$
– Wang
Dec 23 '18 at 2:49