On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle

Abstract: Let $0<κ<1$ and set $p_+=2/(1-κ)$ and $p_-=2/(1+κ)$. We construct a coefficient $κ\leq a\leqκ<sup>{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W<sup>{1,2}<em>{p</em>+}$ 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 $W<sup>{1,2}<em>{p</em>-}$ a priori estimate also fails. Conversely, we demonstrate that the $W<sup>{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|&lt;cκ$, 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 $$p\&gt;2-cκ$$ and this order is sharp as $κ\to 0$. The corresponding results for divergence form equations are also obtained.