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

Abstract

Let $0<\kappa<1$ and set $p_+=2/(1-\kappa)$ and $p_-=2/(1+\kappa)$. We construct a coefficient $\kappa\leq a\leq\kappa^{-1}$, 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 $|p-2|<c\kappa$, showing that the size of the solvability interval around $2$ has optimal order $\kappa$ as $\kappa\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-c\kappa$ and this order is sharp as $\kappa\to 0$. The corresponding results for divergence form equations are also obtained.