Proof of Hardy inequality in $mathbbR^n$
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I've often seen people use the inequality
$$int_mathbb R^3 frac^2,dx leq 4int_mathbb R^3|nabla u(x)|^2,dx,qquad uin C_0^infty(mathbb R^3) $$
without proof, refering to it as "Hardy's inequality".
I struggled to find a direct proof of this in the literature and couldn't prove it myself. Does anyone know a straightforward proof of this or a book in which Hardy's inequality in this form is proved?
I would also be interested in the general form of this inequality, i.e. what happens if one replaces $mathbb R^3$ with $mathbb R^n$?
functional-analysis inequality lebesgue-integral sobolev-spaces lp-spaces
asked Sep 6 at 10:32
Frank
17213
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up vote
3
down vote
favorite
I've often seen people use the inequality
$$int_mathbb R^3 frac^2,dx leq 4int_mathbb R^3|nabla u(x)|^2,dx,qquad uin C_0^infty(mathbb R^3) $$
without proof, refering to it as "Hardy's inequality".
I struggled to find a direct proof of this in the literature and couldn't prove it myself. Does anyone know a straightforward proof of this or a book in which Hardy's inequality in this form is proved?
I would also be interested in the general form of this inequality, i.e. what happens if one replaces $mathbb R^3$ with $mathbb R^n$?
functional-analysis inequality lebesgue-integral sobolev-spaces lp-spaces
asked Sep 6 at 10:32
Frank
17213
Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I've often seen people use the inequality
$$int_mathbb R^3 frac^2,dx leq 4int_mathbb R^3|nabla u(x)|^2,dx,qquad uin C_0^infty(mathbb R^3) $$
without proof, refering to it as "Hardy's inequality".
I struggled to find a direct proof of this in the literature and couldn't prove it myself. Does anyone know a straightforward proof of this or a book in which Hardy's inequality in this form is proved?
I would also be interested in the general form of this inequality, i.e. what happens if one replaces $mathbb R^3$ with $mathbb R^n$?
functional-analysis inequality lebesgue-integral sobolev-spaces lp-spaces
asked Sep 6 at 10:32
Frank
17213
I've often seen people use the inequality
$$int_mathbb R^3 frac^2,dx leq 4int_mathbb R^3|nabla u(x)|^2,dx,qquad uin C_0^infty(mathbb R^3) $$
without proof, refering to it as "Hardy's inequality".
I struggled to find a direct proof of this in the literature and couldn't prove it myself. Does anyone know a straightforward proof of this or a book in which Hardy's inequality in this form is proved?
I would also be interested in the general form of this inequality, i.e. what happens if one replaces $mathbb R^3$ with $mathbb R^n$?
functional-analysis inequality lebesgue-integral sobolev-spaces lp-spaces
functional-analysis inequality lebesgue-integral sobolev-spaces lp-spaces
asked Sep 6 at 10:32
Frank
17213
asked Sep 6 at 10:32
Frank
17213
asked Sep 6 at 10:32
Frank
17213
asked Sep 6 at 10:32
Frank
17213
asked Sep 6 at 10:32
Frank
17213
17213
Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58
add a comment |Â
Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58
Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,
Hardy's inequality- A technical step in proving Hardy's inequality
However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.
answered Sep 6 at 11:01
Calvin Khor
8,84621133
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,
Hardy's inequality- A technical step in proving Hardy's inequality
However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.
answered Sep 6 at 11:01
Calvin Khor
8,84621133
add a comment |Â
up vote
3
down vote
accepted
This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,
Hardy's inequality- A technical step in proving Hardy's inequality
However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.
answered Sep 6 at 11:01
Calvin Khor
8,84621133
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,
Hardy's inequality- A technical step in proving Hardy's inequality
However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.
answered Sep 6 at 11:01
Calvin Khor
8,84621133
This is in Evans' PDE book, page 296 in the second edition. The following discussions on this proof may also be useful,
Hardy's inequality- A technical step in proving Hardy's inequality
However, it doesn't seem like the constant 4 is explicitly computed in Evans, and moreover it may be improved if the domain of $u$ is not convex. In the dimension 1 variant, it is optimal, and you can refer to Computing the best constant in classical Hardy's inequality. In higher dimensions, you may want to look at this paper On the best constant for Hardy's inequality in $mathbb R^n$ by Marcus, Mizel and Pinchover, and its references.
answered Sep 6 at 11:01
Calvin Khor
8,84621133
edited Sep 6 at 11:14
answered Sep 6 at 11:01
Calvin Khor
8,84621133
answered Sep 6 at 11:01
Calvin Khor
8,84621133
answered Sep 6 at 11:01
Calvin Khor
8,84621133
8,84621133
add a comment |Â
add a comment |Â
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Have you seen Evans? Its page 296 in the second edition
– Calvin Khor
Sep 6 at 10:48
Oh, thank you so much! I only looked in the Index of Evans' book and there was no entry for "Hardy Inequality". This is exactly what I was looking for
– Frank
Sep 6 at 10:58