发表机构
Brown University; City University of Hong Kong（布朗大学; 香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一维可测系数抛物方程，构造反例证明Sobolev估计在临界指数处失效，并证明在2附近最优阶为κ的区间内估计成立且唯一可解，进而得到最优的Alexandrov极大值原理。
AI 中文摘要
设$0<\kappa<1$，并令$p_+=2/(1-\kappa)$和$p_-=2/(1+\kappa)$。我们构造一个系数$\kappa\leq a\leq\kappa^{-1}$，该系数在Lebesgue测度为零的紧集外部光滑，对于该系数，一维非散度型抛物方程的$W^{1,2}_{p_+}$估计不成立。相应的解具有弱$L_{p_+}$空间中的二阶空间导数，但不在$L_{p_+}$中。通过对偶性，$W^{1,2}_{p_-}$先验估计也不成立。相反，我们证明了对于$|p-2|<c\kappa$，$W^{1,2}_p$估计成立且方程唯一可解，这表明当$\kappa\downarrow0$时，围绕2的可解性区间的大小具有最优阶$\kappa$，这解决了文献[17]中提出的一个问题。这些结果意味着Alexandrov极大值原理对于$p>2-c\kappa$成立，并且当$\kappa\to 0$时该阶是最优的。散度型方程的相应结果也已获得。
英文摘要
Let $0<κ<1$ and set $p_+=2/(1-κ)$ and $p_-=2/(1+κ)$. We construct a coefficient $κ\leq a\leqκ^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|<cκ$, showing that the size of the solvability interval around $2$ has optimal order $κ$ as $κ\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-cκ$ and this order is sharp as $κ\to 0$. The corresponding results for divergence form equations are also obtained.
