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
asked Dec 27 '18 at 17:37
sharpe
26312
This question has an open bounty worth +50
reputation from sharpe ending in 7 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
asked Dec 27 '18 at 17:37
sharpe
26312
This question has an open bounty worth +50
reputation from sharpe ending in 7 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
asked Dec 27 '18 at 17:37
sharpe
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
calculus
asked Dec 27 '18 at 17:37
sharpe
26312
asked Dec 27 '18 at 17:37
sharpe
26312
edited Dec 27 '18 at 17:48
asked Dec 27 '18 at 17:37
sharpe
26312
asked Dec 27 '18 at 17:37
sharpe
26312
asked Dec 27 '18 at 17:37
sharpe
26312
26312
This question has an open bounty worth +50
reputation from sharpe ending in 7 days.
This question has not received enough attention.
This question has an open bounty worth +50
reputation from sharpe ending in 7 days.
This question has not received enough attention.
add a comment |
add a comment |
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