Mixing
Harmonic functions with a boundary condition.
$begingroup$
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}}begin{bmatrix}2x\-{zover x^2}end{bmatrix}quad,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 $Delta u=({partial^2 overpartial x_1^2}+cdots{partial^2 overpartial x_d^2})u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
asked Dec 27 '18 at 17:37
sharpesharpe
26312
$endgroup$
add a comment |
$begingroup$
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}}begin{bmatrix}2x\-{zover x^2}end{bmatrix}quad,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 $Delta u=({partial^2 overpartial x_1^2}+cdots{partial^2 overpartial x_d^2})u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
asked Dec 27 '18 at 17:37
sharpesharpe
26312
$endgroup$
$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43
add a comment |
$begingroup$
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}}begin{bmatrix}2x\-{zover x^2}end{bmatrix}quad,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 $Delta u=({partial^2 overpartial x_1^2}+cdots{partial^2 overpartial x_d^2})u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
asked Dec 27 '18 at 17:37
sharpesharpe
26312
$endgroup$
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}}begin{bmatrix}2x\-{zover x^2}end{bmatrix}quad,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 $Delta u=({partial^2 overpartial x_1^2}+cdots{partial^2 overpartial x_d^2})u=0$ and $(nabla u,nu)=0$.
calculus harmonic-functions
calculus harmonic-functions
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
asked Dec 27 '18 at 17:37
sharpesharpe
26312
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
asked Dec 27 '18 at 17:37
sharpesharpe
26312
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
edited Jan 8 at 16:43
Mostafa Ayaz
15.4k3939
15.4k3939
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
$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43
add a comment |
$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43
$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43
add a comment |
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$begingroup$
What is $d$ here? And what is meant by find, is it existence or do you hope to get a closed form solution?
$endgroup$
– Andrew
Jan 6 at 8:04
$begingroup$
Sorry, $d=3$. I want to know a closed form solution. Nontrivial harmonic functions closely relate to stochastic processes on $D$.
$endgroup$
– sharpe
Jan 7 at 6:02
$begingroup$
The boundary is not smooth. Is there a reason to expect a closed form solution?
$endgroup$
– Andrew
Jan 7 at 11:43