How can the method of spherical means be used to prove uniqueness of the wave equation?
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I heard that the method of spherical means can be used to prove uniqueness of the wave equation solution for the initial value problem of a wave at time $0$ and velocity $0$. I've found almost no PDFs about this method. The only good one is this, however it does not prove unicity.
I believe the proof is simple, but it's not something I'd come with because I don't have the intuition on why this method works.
real-analysis pde wave-equation
asked Nov 20 at 2:03
Lucas Zanella
86411330
add a comment |
up vote
1
down vote
favorite
I heard that the method of spherical means can be used to prove uniqueness of the wave equation solution for the initial value problem of a wave at time $0$ and velocity $0$. I've found almost no PDFs about this method. The only good one is this, however it does not prove unicity.
I believe the proof is simple, but it's not something I'd come with because I don't have the intuition on why this method works.
real-analysis pde wave-equation
asked Nov 20 at 2:03
Lucas Zanella
86411330
1
A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I heard that the method of spherical means can be used to prove uniqueness of the wave equation solution for the initial value problem of a wave at time $0$ and velocity $0$. I've found almost no PDFs about this method. The only good one is this, however it does not prove unicity.
I believe the proof is simple, but it's not something I'd come with because I don't have the intuition on why this method works.
real-analysis pde wave-equation
asked Nov 20 at 2:03
Lucas Zanella
86411330
I heard that the method of spherical means can be used to prove uniqueness of the wave equation solution for the initial value problem of a wave at time $0$ and velocity $0$. I've found almost no PDFs about this method. The only good one is this, however it does not prove unicity.
I believe the proof is simple, but it's not something I'd come with because I don't have the intuition on why this method works.
real-analysis pde wave-equation
real-analysis pde wave-equation
asked Nov 20 at 2:03
Lucas Zanella
86411330
asked Nov 20 at 2:03
Lucas Zanella
86411330
asked Nov 20 at 2:03
Lucas Zanella
86411330
asked Nov 20 at 2:03
Lucas Zanella
86411330
asked Nov 20 at 2:03
Lucas Zanella
86411330
86411330
1
A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46
add a comment |
1
A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46
1
1
A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46
add a comment |
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A detailed exposition of the wave equation in $mathbb{R}^n$ can be found in Evan's PDE book. He uses the same method to derive Kirchhoff's formula (for even and odd $n$. I don't necessarily think its traditional to prove uniqueness via those means, because of the efficiency of the energy method to do so.
– DaveNine
Nov 20 at 9:43
For the energy method, we suppose $$e(t) = int_U w_t^2(x,t)+|Dw(x,t)|^2dx$$ where $w = u - bar{u}$ is the difference between two solutions to the initial value problem for the wave equation, then show that $e(t) = 0$.
– DaveNine
Nov 20 at 9:46