Eigenfunctions for Laplacian on increasing domains
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I was wondering if we can compare the eigenfunctions of the Laplacian for increasing domains. In particular, I want to look at the eigenfunction for the principal eigenvalue with integral $1$. Is it possible that we could get pointwise boundedness? I have seen that the eigenvalues themselves decrease as the domains increase.
real-analysis functional-analysis analysis pde
asked Nov 20 at 0:33
David Bowman
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I was wondering if we can compare the eigenfunctions of the Laplacian for increasing domains. In particular, I want to look at the eigenfunction for the principal eigenvalue with integral $1$. Is it possible that we could get pointwise boundedness? I have seen that the eigenvalues themselves decrease as the domains increase.
real-analysis functional-analysis analysis pde
asked Nov 20 at 0:33
David Bowman
4,2391924
I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I was wondering if we can compare the eigenfunctions of the Laplacian for increasing domains. In particular, I want to look at the eigenfunction for the principal eigenvalue with integral $1$. Is it possible that we could get pointwise boundedness? I have seen that the eigenvalues themselves decrease as the domains increase.
real-analysis functional-analysis analysis pde
asked Nov 20 at 0:33
David Bowman
4,2391924
I was wondering if we can compare the eigenfunctions of the Laplacian for increasing domains. In particular, I want to look at the eigenfunction for the principal eigenvalue with integral $1$. Is it possible that we could get pointwise boundedness? I have seen that the eigenvalues themselves decrease as the domains increase.
real-analysis functional-analysis analysis pde
real-analysis functional-analysis analysis pde
asked Nov 20 at 0:33
David Bowman
4,2391924
asked Nov 20 at 0:33
David Bowman
4,2391924
asked Nov 20 at 0:33
David Bowman
4,2391924
asked Nov 20 at 0:33
David Bowman
4,2391924
asked Nov 20 at 0:33
David Bowman
4,2391924
4,2391924
I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46
add a comment |
I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46
I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46
add a comment |
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I have a result in my dissertation that is relevant to this question!! (Never thought I'd type that sentence.) It's not exactly your question, hence not an answer -- but maybe still interesting. If you normalize by volume and make an assumption about the boundary saying the domains don't stretch out too thin, then the eigenvalues limit to the value predicted by Weyl's law. However if you don't make the assumption, then there are easy counterexamples. This is Proposition 4.13 in arxiv 1808.07206
– Neal
Nov 20 at 1:40
More directly to the question actually asked -- if you want to precisely define a question about pointwise boundedness, you have to pull back the eigenfunctions to a single domain, so that you're no longer studying varying domains, but a single domain with a varying Riemannian metric.
– Neal
Nov 20 at 1:46