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Small-scale mass estimates for Neumann eigenfunctions: piecewise smooth planar domains
Authors
Hans Christianson
John A. Toth
Publication date
19 September 2023
Publisher
View
on
arXiv
Abstract
Let
Ω
\Omega
Ω
be a piecewise-smooth, bounded convex domain in
R
2
\R^2
R
2
and consider
L
2
L^2
L
2
-normalized Neumann eigenfunctions
Ï•
λ
\phi_{\lambda}
Ï•
λ
​
with eigenvalue
λ
2
\lambda^2
λ
2
. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any}
x
0
∈
Ω
‾
,
x_0 \in \overline{\Omega},
x
0
​
∈
Ω
,
(including boundary and corner points) and any
δ
∈
[
0
,
1
)
,
\delta \in [0,1),
δ
∈
[
0
,
1
)
,
∥
Ï•
λ
∥
B
(
x
0
,
λ
−
δ
)
∩
Ω
=
O
(
λ
−
δ
/
2
)
.
\| \phi_\lambda \|_{B(x_0,\lambda^{-\delta})\cap \Omega} = O(\lambda^{-\delta/2}).
∥
Ï•
λ
​
∥
B
(
x
0
​
,
λ
−
δ
)
∩
Ω
​
=
O
(
λ
−
δ
/2
)
.
The proof is a stationary vector field argument combined with a small scale induction argument.Comment: This is an expanded version of the first half of the preprint arXiv:2012.15237 [math.AP] by the same author
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oai:arXiv.org:2309.10875
Last time updated on 10/10/2023