Received 1 June 2016; accepted 24 June 2016; published 27 June 2016
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  -  , 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
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
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
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
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
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.
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.
 Han, Q. and Lin, F. (2011) Elliptic Partial Differential Equations. 2rd Edition, Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence.
 Moser, J. (1964) A Harnack Inequality for Parabolic Differential Equations. Communications on Pure and Applied Mathematics, 17, 101-134.
 Moser, J. (1961) On Harnack’s Theorem for Elliptic Differential Equations. Communications on Pure and Applied Mathematics, 14, 577-591.