In this paper, we consider L∞ estimates of eigenfunction, or more generally, the L∞ estimates of equation -Δu=fu. We use heat flow to give a new proof of the L∞ estimates for such type equations.
Let
where
equation. If
be the heat kernel in
where
Theorem 1. Let
on
where
Remark 2. Following from the proof, one can consider equation
To estimates on
Lemma 1. Let
on
Proof. Let
By the heat equation
where we use integrating by parts for term
Hence, for
Since
By the property of heat kernel, we have
On the other hand, as
Combining with
Hence we finish the proof.
The following lemma is fundamental.
Lemma 2. For any
Proof. Let
Now we are ready to prove Theorem 1.
Proof of Theorem 1. Refmaintheorem. For any compact subset
on
If
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.
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