Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations
Uniform restricted parabolic Harnack inequality, separation principle, and ultracontractivity for parabolic equations
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抛物方程的均匀受限抛物线 Harnack 不等式、分离原理和超收缩性
DOI：
10.1007/bfb0085486
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
M. Murata
中科院分区：
文献类型：
--
作者：
M. Murata
A. Freire [F]利用P. Li & ST Yau [LY]的抛物Harnack不等式建立了黎曼积上正调和函数的一个漂亮的分裂定理。受他的工作和随后的一篇论文的启发，Taylor [T]，本文引入了Rn+1柱上二阶线性抛物型方程（at+ L）u= 0非负解的一致限制抛物Harnack不等式（URPH）的概念，其中L是时间无关的椭圆算子.本文的目的是讨论与URPH有关的几个性质。在§ 1中，我们建立（见下面的定理1.3）URPH等价于分离原理（粗略地说，抛物方程的非负解是最小的当且仅当它是eAt和椭圆方程的最小正解的乘积）。我们还证明了（见下面的定理1.4）URPH蕴含正柯西问题的唯一性，在§ 2（见下面的定理2.2）中，URPH给出了椭圆方程正解的分裂定理。在§ 3中，我们建立了（见下面的定理3.1）薛定谔半群（d. [DS])意味着正柯西问题的非唯一性;因此，如果IU成立，则URPH不成立。这个非唯一性定理是独立的兴趣。在§§ 1””3中我们处理R”x I中的抛物方程，其中I是开区间，而在§ 4中我们处理一般圆柱上的方程。
A. Freire [F] made use of a parabolic Harnack inequality due to P. Li & ST Yau [LY] in establishing a beautiful splitting theorem for positive harmonic functions on a Riemannian product. Inspired by his work and a subsequent paper by J. C. Taylor [T], we introduce in this paper the notion of uniform restricted parabolic Harnack inequality (URPH) for nonnegative solutions of a linear second order parabolic equation (at+ L) u= 0 on a cylinder of Rn+ l, where L is a time independent elliptic operator. The purpose of this paper is to discuss several properties related to URPH. In § 1, we establish (see Theorem 1.3 below) that URPH is equivalent to the separation principle (which says, roughly speaking, that a nonnegative solution of a parabolic equation is minimal iff it is a product of eAt and a minimal positive solution of an elliptic equation). We also show there (see Theorem 1.4 below) that URPH implies uniqueness of the positive Cauchy problem, and in § 2 (see Theorem 2.2 below) that URPH yields a splitting theorem for positive solutions of an elliptic equation. In § 3 we establish (see Theorem 3.1 below) that the intrinsic ultracontractivity (IU) of a Schrodinger semigroup (d.[DS]) implies nonuniqueness of the positive Cauchy problem; therfore, if IU holds, then URPH does not hold. This nonuniqueness theorem is of independent interest. While we treat in §§ 1"" 3 parabolic equations in R" x I, where I is an open interval, we do in § 4 equations in general cylinders.