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

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