De Silva和Savin高阶比较定理的抛物型类似
A parabolic analogue of the higher-order comparison theorem of De Silva and Savin
摘要：
我们证明了在$H^{k+ α}$域的侧边界的一部分上消失的两个热函数的商是$H^{k+ α}$直到$k \geq 2$的边界。在$k=1$的情况下，我们表明，商是在$H^{1+α}$，如果域被假定为时空$C^{1, α}$正则。这可以被认为是一个抛物线模拟最近的一个重要结果[DS1]，我们密切关注的想法在该文件中。我们还给出了反例，说明类似的结果在抛物边界上的点上不成立，而这些点不在侧边界上，即，在抛物线边界的拐角和底部的点。
英文摘要：
We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ α}$ domain is $H^{k+ α}$ up to the boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+α}$ if the domain is assumed to be space-time $C^{1, α}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DS1], and we closely follow the ideas in that paper. We also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary.
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