A new proof of Moser's parabolic harnack inequality using the old ideas of Nash
A new proof of Moser's parabolic harnack inequality using the old ideas of Nash
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DOI：
10.1007/bf00251802
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发表时间：
1986-12
影响因子：
2.5
通讯作者：
E. Fabes;E. Fabes;D. Stroock;D. Stroock
中科院分区：
文献类型：
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作者：
E. Fabes;E. Fabes;D. Stroock;D. Stroock
纳什在1958年出版了他的基本工作的地方保持器连续性的解决方案的二阶抛物方程的非光滑系数（[7]）。这项工作的主要目的是研究基本解的性质对应的抛物算子，并从这些性质的正则性一般的解决方案。虽然这一工作经常在关于椭圆和抛物方程弱解的文献中被引用，但人们认为纳什的思想从未被完全理解（也许现在仍然没有），因此DeGiorgi（[3]）和Moser（[5]，[6]）的更容易理解和似乎更有成果的思想随后被采用。特别是，通过修改和说服他的论点，我们直接建立了我们认为是这条推理路线的逻辑目标，即：DG Aronson（[1]）首先证明的基本解的估计。根据Aronson对Moser（[6]）的抛物Harnack不等式的估计，从而（如Moser [6，p.108]所示），抛物方程弱解的Nash局部H6lder连续性成立。也就是说，我们的方法颠倒了这些结果最初得出的时间顺序。为了使上述陈述在数学上精确，我们介绍了贯穿本工作的基本符号和定义。我们将学习以下形式的抛物算子：
In 1958 Nash published his fundamental work on the local Holder continuity of solutions of second order parabolic equations with non-smooth coefficients ([7]). The primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and to derive from these properties the regularity for a general solution. Though the work is often cited in the literature about weak solutions of elliptic and parabolic equations, one feels that Nash's ideas were never fully understood (and maybe still are not) and that because of this the more understandable and seemingly more fruitful ideas of DeGiorgi ([3]) and Moser ([5],[6]) were subsequently adopted.In the present article, we return to Nash's ideas. In particular, by modifying and persuing his arguments, we establish directly what we feel is the logical goal of this line of reasoning, namely: the estiamtes for the fundamental solution first proved by DG Aronson ([1]). From Aronson's estimates the parabolic Harnack inequality of Moser ([6]) and, consequently (as was shown by Moser [6, p. 108]), Nash's local H6lder continuity of weak solutions to parabolic equations follow. That is, our approach reverses the chronological order in* which these results were derived originally. To make the above statements mathematically precise we introduce the basic notations and definitions to be used throughout this work. We will be studying parabolic operators of the form