Harmonic functions with a boundary condition.
I am looking for a harmonic function.
Let $H(x)=x^2$ and
let $D={(x,z) in mathbb{R} times mathbb{R}^2 mid x>1, |z|<H(x)}$.
Here, $|cdot|$ denotes the $2$-dim Euclid norm.
$D$ is an unbounded domain of $mathbb{R}^3$.
The inward normal unit vector $nu$ on $partial D$ is expressed as
begin{equation*}
nu(x,z)=frac{1}{(4x^2+1)^{1/2}}(2x,-z/x^2),quad |z|=x^2.
end{equation*}
We can easily find a nontrivial smooth function $u$ on $D$ with Neumann boundary condition: $(nabla u,nu)=0$ on $partial D$. Here, $nabla u$ denotes the gradient of $u$ and $(cdot,cdot)$ denotes the standard inner product.
My question
Can we find a nontrivial harmonic function $u$ on $D$ with Neumann boundary condition? Namely,$u$ satisfies $Delta u=(partial^2 /partial x_1^2+cdots+partial^2 /partial x_d^2)u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited 2 hours ago
El Pasta
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
This question has an open bounty worth +50
reputation from sharpe ending in 3 days.
This question has not received enough attention.
add a comment |
I am looking for a harmonic function.
Let $H(x)=x^2$ and
let $D={(x,z) in mathbb{R} times mathbb{R}^2 mid x>1, |z|<H(x)}$.
Here, $|cdot|$ denotes the $2$-dim Euclid norm.
$D$ is an unbounded domain of $mathbb{R}^3$.
The inward normal unit vector $nu$ on $partial D$ is expressed as
begin{equation*}
nu(x,z)=frac{1}{(4x^2+1)^{1/2}}(2x,-z/x^2),quad |z|=x^2.
end{equation*}
We can easily find a nontrivial smooth function $u$ on $D$ with Neumann boundary condition: $(nabla u,nu)=0$ on $partial D$. Here, $nabla u$ denotes the gradient of $u$ and $(cdot,cdot)$ denotes the standard inner product.
My question
Can we find a nontrivial harmonic function $u$ on $D$ with Neumann boundary condition? Namely,$u$ satisfies $Delta u=(partial^2 /partial x_1^2+cdots+partial^2 /partial x_d^2)u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited 2 hours ago
El Pasta
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
This question has an open bounty worth +50
reputation from sharpe ending in 3 days.
This question has not received enough attention.
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago
add a comment |
I am looking for a harmonic function.
Let $H(x)=x^2$ and
let $D={(x,z) in mathbb{R} times mathbb{R}^2 mid x>1, |z|<H(x)}$.
Here, $|cdot|$ denotes the $2$-dim Euclid norm.
$D$ is an unbounded domain of $mathbb{R}^3$.
The inward normal unit vector $nu$ on $partial D$ is expressed as
begin{equation*}
nu(x,z)=frac{1}{(4x^2+1)^{1/2}}(2x,-z/x^2),quad |z|=x^2.
end{equation*}
We can easily find a nontrivial smooth function $u$ on $D$ with Neumann boundary condition: $(nabla u,nu)=0$ on $partial D$. Here, $nabla u$ denotes the gradient of $u$ and $(cdot,cdot)$ denotes the standard inner product.
My question
Can we find a nontrivial harmonic function $u$ on $D$ with Neumann boundary condition? Namely,$u$ satisfies $Delta u=(partial^2 /partial x_1^2+cdots+partial^2 /partial x_d^2)u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited 2 hours ago
El Pasta
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
I am looking for a harmonic function.
Let $H(x)=x^2$ and
let $D={(x,z) in mathbb{R} times mathbb{R}^2 mid x>1, |z|<H(x)}$.
Here, $|cdot|$ denotes the $2$-dim Euclid norm.
$D$ is an unbounded domain of $mathbb{R}^3$.
The inward normal unit vector $nu$ on $partial D$ is expressed as
begin{equation*}
nu(x,z)=frac{1}{(4x^2+1)^{1/2}}(2x,-z/x^2),quad |z|=x^2.
end{equation*}
We can easily find a nontrivial smooth function $u$ on $D$ with Neumann boundary condition: $(nabla u,nu)=0$ on $partial D$. Here, $nabla u$ denotes the gradient of $u$ and $(cdot,cdot)$ denotes the standard inner product.
My question
Can we find a nontrivial harmonic function $u$ on $D$ with Neumann boundary condition? Namely,$u$ satisfies $Delta u=(partial^2 /partial x_1^2+cdots+partial^2 /partial x_d^2)u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
calculus harmonic-functions
edited 2 hours ago
El Pasta
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
edited 2 hours ago
El Pasta
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
edited 2 hours ago
El Pasta
17714
edited 2 hours ago
El Pasta
17714
edited 2 hours ago
El Pasta
17714
17714
asked Dec 27 '18 at 17:37
sharpesharpe
26312
asked Dec 27 '18 at 17:37
sharpesharpe
26312
asked Dec 27 '18 at 17:37
sharpesharpe
26312
26312
This question has an open bounty worth +50
reputation from sharpe ending in 3 days.
This question has not received enough attention.
This question has an open bounty worth +50
reputation from sharpe ending in 3 days.
This question has not received enough attention.
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago
add a comment |
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago
add a comment |
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What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
– Andrew
yesterday
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
– sharpe
20 hours ago
The boundary is not smooth. Is there a reason to expect a closed form solution?
– Andrew
14 hours ago