Advances in Pure Mathematics
Vol.06 No.07(2016), Article ID:67713,4 pages
10.4236/apm.2016.67038
A Remark on Eigenfunction Estimates by Heat Flow
Huabin Ge1, Yipeng Shi2
1Department of Mathematics, Beijing Jiaotong University, Beijing, China
2College of Engineering, Peking University, Beijing, China
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 1 June 2016; accepted 24 June 2016; published 27 June 2016
ABSTRACT
In this paper, we consider estimates of eigenfunction, or more generally, the
estimates of equation
. We use heat flow to give a new proof of the
estimates for such type equations.
Keywords:
Estimates, Eigenfunction, Heat Flow
1. Introduction
Let be a bounded domain. Assume
, we consider the Laplacian equation
where and
with
. This is a second order differential
equation. If is a constant, then u is an eigenfunction with eigenvalue
. By a standard Moser’s iteration in [1] - [5] , we have
interior estimates of u controlled by the
norm of u for
. In this paper, we use heat flow to consider the
estimate and give a new proof of the
estimates without using iteration. First, we recall the definition of the heat kernel. For any
and
, let
be the heat kernel in. For fixed
, we know that
where is the standard Laplacian in
with respect to x. Our main result is the following
Theorem 1. Let be a bounded domain with
. Assume
and
on with
. Then for any
and any compact sub-domain
, we have the interior
estimate
(1)
where is the distance of
and the boundary of
.
Remark 2. Following from the proof, one can consider equation or
by choosing appropriate kernel function
.
2. Proving the Theorem
To estimates on, by the translation invariant and scaling invariant of the estimates, we only need to consider
and
. By using heat flow, we have the following lemma.
Lemma 1. Let be a unite ball. Assume
and
on with
. Then for any
, we have the interior
estimate
(2)
Proof. Let be a standard smooth cutoff function with support in
and
on
, moreover,
. For any
, let
By the heat equation, integrating by parts, we have
(3)
(4)
(5)
(6)
(7)
(8)
where we use integrating by parts for term to get (7) from (6). By direct estimate, since
for
and
, then
. Therefore, for
, we have
Hence, for and noting that
, we have
Since, then we have
By the property of heat kernel, we have. Then we have
On the other hand, as, we have
(9)
Combining with, we have
Hence we finish the proof.
The following lemma is fundamental.
Lemma 2. For any and any
, we have
Proof. Let and
. Then
(10)
(11)
Now we are ready to prove Theorem 1.
Proof of Theorem 1. Refmaintheorem. For any compact subset, let
. For any
, we have
. Consider equation
on. By Lemma 1, since the estimates are scaling invariant, we have
If, then
. By Lemma 2, we have
Hence we finish the proof.
Acknowledgements
The research is supported by National Natural Science Foundation of China under grant No.11501027. The first author would like to thank Dr. Wenshuai Jiang, Xu Xu for many helpful conversations.
Cite this paper
Huabin Ge,Yipeng Shi, (2016) A Remark on Eigenfunction Estimates by Heat Flow. Advances in Pure Mathematics,06,512-515. doi: 10.4236/apm.2016.67038
References
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