Received 1 June 2016; accepted 24 June 2016; published 27 June 2016
![](//html.scirp.org/file/4-5301127x10.png)
1. Introduction
Let
be a bounded domain. Assume
, we consider the Laplacian equation
![](//html.scirp.org/file/4-5301127x13.png)
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.