Elliptic regularity on compact manifold without boundary
up vote
4
down vote
favorite
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $uin H^1(M)$, and $fin L^2(M)$ such that $Delta u = f$ (in the sens of distributions), Then $u in H^2(M)$.
If there is a nice reference for such regularity result It would be good.
reference-request riemannian-geometry elliptic-pde manifolds regularity
asked Dec 6 at 23:13
S. Cho
1728
add a comment |
up vote
4
down vote
favorite
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $uin H^1(M)$, and $fin L^2(M)$ such that $Delta u = f$ (in the sens of distributions), Then $u in H^2(M)$.
If there is a nice reference for such regularity result It would be good.
reference-request riemannian-geometry elliptic-pde manifolds regularity
asked Dec 6 at 23:13
S. Cho
1728
1
I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
1
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $uin H^1(M)$, and $fin L^2(M)$ such that $Delta u = f$ (in the sens of distributions), Then $u in H^2(M)$.
If there is a nice reference for such regularity result It would be good.
reference-request riemannian-geometry elliptic-pde manifolds regularity
asked Dec 6 at 23:13
S. Cho
1728
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $uin H^1(M)$, and $fin L^2(M)$ such that $Delta u = f$ (in the sens of distributions), Then $u in H^2(M)$.
If there is a nice reference for such regularity result It would be good.
reference-request riemannian-geometry elliptic-pde manifolds regularity
reference-request riemannian-geometry elliptic-pde manifolds regularity
asked Dec 6 at 23:13
S. Cho
1728
asked Dec 6 at 23:13
S. Cho
1728
asked Dec 6 at 23:13
S. Cho
1728
asked Dec 6 at 23:13
S. Cho
1728
asked Dec 6 at 23:13
S. Cho
1728
1728
1
I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
1
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35
add a comment |
1
I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
1
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35
1
1
I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
1
1
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35
add a comment |
3 Answers
3
active
oldest
votes
up vote
7
down vote
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $Omega subset mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $Omega$,
$$ tag{*} |u|_{H^2} le C|Delta_0 u|_{L^2}, $$
where $Delta_0$ is the standard flat Laplacian.
To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$
answered Dec 7 at 2:24
Deane Yang
20k562140
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
add a comment |
up vote
6
down vote
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
add a comment |
up vote
0
down vote
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $rho=1, delta=0$.
answered 2 days ago
mcd
411310
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $Omega subset mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $Omega$,
$$ tag{*} |u|_{H^2} le C|Delta_0 u|_{L^2}, $$
where $Delta_0$ is the standard flat Laplacian.
To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$
answered Dec 7 at 2:24
Deane Yang
20k562140
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
add a comment |
up vote
7
down vote
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $Omega subset mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $Omega$,
$$ tag{*} |u|_{H^2} le C|Delta_0 u|_{L^2}, $$
where $Delta_0$ is the standard flat Laplacian.
To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$
answered Dec 7 at 2:24
Deane Yang
20k562140
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
add a comment |
up vote
7
down vote
up vote
7
down vote
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $Omega subset mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $Omega$,
$$ tag{*} |u|_{H^2} le C|Delta_0 u|_{L^2}, $$
where $Delta_0$ is the standard flat Laplacian.
To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$
answered Dec 7 at 2:24
Deane Yang
20k562140
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $Omega subset mathbb{R}^n$, there exists $C>0$ such that for any function (or even just a distribution) $u$ compactly supported in $Omega$,
$$ tag{*} |u|_{H^2} le C|Delta_0 u|_{L^2}, $$
where $Delta_0$ is the standard flat Laplacian.
To extend this to a local regularity estimate for the Laplace-Beltrami operator, it suffices to prove regularity estimate for $u$ compactly supported on a neighborhood of each point $p in M$. If you use geodesic normal coordinates on a sufficiently small neighborhood of $p$, then you can assume that the Laplace-Beltrami operator is of the form
$$
Delta u = (delta^{ij} + a^{ij}(x))partial^2 + b^kpartial_ku
$$
where $|a^{ij}|, |b_k| < epsilon << 1$.
Therefore, if $Delta_g u = f$, then
$$
Delta_0u = -a_{ij}partial^2_{ij}u - b^kpartial_ku + f
$$
Therefore, by $(*)$
$$
|u|_{H^2} le C(epsilon |u|_{H^2} + |f|_{L^2}).
$$
If the neighborhood is sufficiently small, then $Cepsilon < 1$ and therefore,
$$
|u|_{H^2} le C|f|_{L^2}.
$$
answered Dec 7 at 2:24
Deane Yang
20k562140
answered Dec 7 at 2:24
Deane Yang
20k562140
answered Dec 7 at 2:24
Deane Yang
20k562140
answered Dec 7 at 2:24
Deane Yang
20k562140
20k562140
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
add a comment |
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
@Yang Thank you ! Can you recommend a good reference for such proof ?
– S. Cho
Dec 7 at 10:32
1
1
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
Unfortunately, I don't know a reference. Lemmas like this are used all the time by PDE people but, since they're used only in very specific circumstances, they rarely appear in books. Roughly the same argument does appear in the appendix of a paper I wrote on convergence of Riemannian manifolds. It's also similar in the spirit to a technique called "freezing coefficients", so you can try searching for books and papers mentioning that.
– Deane Yang
Dec 7 at 16:45
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
There is a similar result in Taylor's book when the $uin H^1_0(M)$. Is this implies the result for my case ?
– S. Cho
Dec 7 at 17:22
1
1
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
I don't know. Note that it does suffice to restrict to functions compactly supported in a bounded open domain. Perhaps you could quote the exact statement of what is in Taylor's book.
– Deane Yang
Dec 7 at 17:39
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
It's Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem.
– S. Cho
Dec 7 at 19:31
add a comment |
up vote
6
down vote
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
add a comment |
up vote
6
down vote
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
add a comment |
up vote
6
down vote
up vote
6
down vote
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
While there are many books that deal with elliptic regularity on manifolds, Warner's book seems most elementary and oriented towards those who do not know much about analysis, but are familiar with geometry of manifolds.
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
answered Dec 7 at 0:17
Piotr Hajlasz
5,92142253
5,92142253
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
add a comment |
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
2
2
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
I am very fond of Wells' "Differential analysis on complex manifolds" chapter on Hodge theory. The approach by way of pseudo-differential operators may feel less elementary, but I think it leads to a clean proof and conceptual insights. I read it when I was an early graduate student who was still getting comfortable with Sobolev spaces.
– Mike Miller
Dec 7 at 2:48
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
@Hajlasz Thank you. Do you mean Theorem 6.30 (Regularity for Periodic Elliptic Operators) since I have an other version of the book.
– S. Cho
Dec 7 at 10:14
2
2
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
@S.Cho I will expand my answer when I am back to the office. Hopefully some time today. I will comment on Warner's proof and add some other references.
– Piotr Hajlasz
Dec 7 at 14:35
3
3
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
@PiotrHajlasz, Warner's book is indeed a wonderful self-contained exposition of important theorems in differential topology, whose proofs are not easily found elsewhere. I also like the way he is able to present proofs of the elliptic PDE theorems needed for the Hodge theory in such a elementary way without the fancy modern machinery.
– Deane Yang
Dec 7 at 16:48
add a comment |
up vote
0
down vote
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $rho=1, delta=0$.
answered 2 days ago
mcd
411310
add a comment |
up vote
0
down vote
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $rho=1, delta=0$.
answered 2 days ago
mcd
411310
add a comment |
up vote
0
down vote
up vote
0
down vote
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $rho=1, delta=0$.
answered 2 days ago
mcd
411310
A more general theorem with a proof using pseudodifferential operators is Theorem 7.2 in Shubin's book (Pseudodifferential operators and Spectral Theory). In your case the operator is second order and elliptic, so $m=m_0 = 2$ and $rho=1, delta=0$.
answered 2 days ago
mcd
411310
answered 2 days ago
mcd
411310
answered 2 days ago
mcd
411310
answered 2 days ago
mcd
411310
411310
add a comment |
add a comment |
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I would look in Partial Differential Equations I and PDE II by Taylor. He develops the theory on manifolds.
– Neal
Dec 7 at 2:26
1
I found Theorem 1.3. in Section 5 : linear elliptic equations, 1: existence and regularity of solutions to the Dirichlet problem. In Taylor I. But I don't know if this implies the desired résult. He assume that $u|_{partial M} =0$, in my case there is no boundary.
– S. Cho
Dec 7 at 19:35