Mixing
Laplacian equation on non-compact manifold
$begingroup$
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold.
For the equation
$$Delta u=f,$$
for some $fin L^2(M)$.
Q How can we find a solution $uin L^2$ satisfies the above equation?
functional-analysis pde laplacian
asked Jan 13 at 5:12
DLINDLIN
406414
$endgroup$
add a comment |
$begingroup$
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold.
For the equation
$$Delta u=f,$$
for some $fin L^2(M)$.
Q How can we find a solution $uin L^2$ satisfies the above equation?
functional-analysis pde laplacian
asked Jan 13 at 5:12
DLINDLIN
406414
$endgroup$
$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45
add a comment |
$begingroup$
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold.
For the equation
$$Delta u=f,$$
for some $fin L^2(M)$.
Q How can we find a solution $uin L^2$ satisfies the above equation?
functional-analysis pde laplacian
asked Jan 13 at 5:12
DLINDLIN
406414
$endgroup$
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold.
For the equation
$$Delta u=f,$$
for some $fin L^2(M)$.
Q How can we find a solution $uin L^2$ satisfies the above equation?
functional-analysis pde laplacian
functional-analysis pde laplacian
asked Jan 13 at 5:12
DLINDLIN
406414
asked Jan 13 at 5:12
DLINDLIN
406414
edited Jan 13 at 5:44
DLIN
asked Jan 13 at 5:12
DLINDLIN
406414
asked Jan 13 at 5:12
DLINDLIN
406414
asked Jan 13 at 5:12
DLINDLIN
406414
406414
$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45
add a comment |
$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45
$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45
$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45
add a comment |
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$begingroup$
In general you can't. In Euclidean space you can use Fourier analysis to arrive at $hat u(xi)=-hat f(xi)/|xi|^2$, which can fail to be square-integrable. Do you have any further conditions?
$endgroup$
– MaoWao
Jan 14 at 15:45