Question on embeddings of $W^{2,p}(partial U)$ space
$begingroup$
I sense that my question might be elementary but I'm quite puzzled about the following:
Let $U subset mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $partial U$. If $f in W^{2,p}(partial U)$ then what is the dimension that I consider in order to use Sobolev embeddings?
Is it true that $n=3$? But how is that possible, since the area of $partial U$ is the $2-$dimensional Hausdorff measure?
Moreover, considering $x_0 in partial U$ then a function $g: B(x_0,r) to mathbb R$ is a function of $2$ or $3$ variables?
I'm pretty sure that I have confused some definitions here so I would appreciate if somebody could help me enlighten this area.
Thank you very much in advance!
functional-analysis differential-geometry pde sobolev-spaces
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
$endgroup$
add a comment |
$begingroup$
I sense that my question might be elementary but I'm quite puzzled about the following:
Let $U subset mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $partial U$. If $f in W^{2,p}(partial U)$ then what is the dimension that I consider in order to use Sobolev embeddings?
Is it true that $n=3$? But how is that possible, since the area of $partial U$ is the $2-$dimensional Hausdorff measure?
Moreover, considering $x_0 in partial U$ then a function $g: B(x_0,r) to mathbb R$ is a function of $2$ or $3$ variables?
I'm pretty sure that I have confused some definitions here so I would appreciate if somebody could help me enlighten this area.
Thank you very much in advance!
functional-analysis differential-geometry pde sobolev-spaces
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
$endgroup$
add a comment |
$begingroup$
I sense that my question might be elementary but I'm quite puzzled about the following:
Let $U subset mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $partial U$. If $f in W^{2,p}(partial U)$ then what is the dimension that I consider in order to use Sobolev embeddings?
Is it true that $n=3$? But how is that possible, since the area of $partial U$ is the $2-$dimensional Hausdorff measure?
Moreover, considering $x_0 in partial U$ then a function $g: B(x_0,r) to mathbb R$ is a function of $2$ or $3$ variables?
I'm pretty sure that I have confused some definitions here so I would appreciate if somebody could help me enlighten this area.
Thank you very much in advance!
functional-analysis differential-geometry pde sobolev-spaces
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
$endgroup$
I sense that my question might be elementary but I'm quite puzzled about the following:
Let $U subset mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $partial U$. If $f in W^{2,p}(partial U)$ then what is the dimension that I consider in order to use Sobolev embeddings?
Is it true that $n=3$? But how is that possible, since the area of $partial U$ is the $2-$dimensional Hausdorff measure?
Moreover, considering $x_0 in partial U$ then a function $g: B(x_0,r) to mathbb R$ is a function of $2$ or $3$ variables?
I'm pretty sure that I have confused some definitions here so I would appreciate if somebody could help me enlighten this area.
Thank you very much in advance!
functional-analysis differential-geometry pde sobolev-spaces
functional-analysis differential-geometry pde sobolev-spaces
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
asked Jan 8 at 11:19
kaithkolesidoukaithkolesidou
1,267512
1,267512
add a comment |
add a comment |
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