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$n$-th Neumann eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1_+$
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It's well know that $n$-th eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1 = x in mathbbR^d$ is generated by harmonics homgeneous polynomial of degree $n$. When we work with the hemisphere $mathbbS^d-1_+ = x = (x_1,dots,x_d) in mathbbR^d$, what subset of set of harmonics homgeneous polynomial of degree $n$ generates $n$-th Neumann eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1_+$ ?
$n$-th Neumann eigenspace of Laplacian operator $Delta$ $:=$ linear space formed by all the eigenfunctions of $Delta$ which satisfy to Neumann boundary condition.
pde riemannian-geometry
asked Jul 25 at 19:45
A.D.
362110
add a comment |Â
up vote
2
down vote
favorite
It's well know that $n$-th eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1 = x in mathbbR^d$ is generated by harmonics homgeneous polynomial of degree $n$. When we work with the hemisphere $mathbbS^d-1_+ = x = (x_1,dots,x_d) in mathbbR^d$, what subset of set of harmonics homgeneous polynomial of degree $n$ generates $n$-th Neumann eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1_+$ ?
$n$-th Neumann eigenspace of Laplacian operator $Delta$ $:=$ linear space formed by all the eigenfunctions of $Delta$ which satisfy to Neumann boundary condition.
pde riemannian-geometry
asked Jul 25 at 19:45
A.D.
362110
Will this article help?
– mvw
Jul 25 at 20:32
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
It's well know that $n$-th eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1 = x in mathbbR^d$ is generated by harmonics homgeneous polynomial of degree $n$. When we work with the hemisphere $mathbbS^d-1_+ = x = (x_1,dots,x_d) in mathbbR^d$, what subset of set of harmonics homgeneous polynomial of degree $n$ generates $n$-th Neumann eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1_+$ ?
$n$-th Neumann eigenspace of Laplacian operator $Delta$ $:=$ linear space formed by all the eigenfunctions of $Delta$ which satisfy to Neumann boundary condition.
pde riemannian-geometry
asked Jul 25 at 19:45
A.D.
362110
It's well know that $n$-th eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1 = x in mathbbR^d$ is generated by harmonics homgeneous polynomial of degree $n$. When we work with the hemisphere $mathbbS^d-1_+ = x = (x_1,dots,x_d) in mathbbR^d$, what subset of set of harmonics homgeneous polynomial of degree $n$ generates $n$-th Neumann eigenspace of Laplacian operator $Delta$ of $mathbbS^d-1_+$ ?
$n$-th Neumann eigenspace of Laplacian operator $Delta$ $:=$ linear space formed by all the eigenfunctions of $Delta$ which satisfy to Neumann boundary condition.
pde riemannian-geometry
asked Jul 25 at 19:45
A.D.
362110
asked Jul 25 at 19:45
A.D.
362110
asked Jul 25 at 19:45
A.D.
362110
asked Jul 25 at 19:45
A.D.
362110
362110
Will this article help?
– mvw
Jul 25 at 20:32
add a comment |Â
Will this article help?
– mvw
Jul 25 at 20:32
Will this article help?
– mvw
Jul 25 at 20:32
Will this article help?
– mvw
Jul 25 at 20:32
add a comment |Â
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Will this article help?
– mvw
Jul 25 at 20:32