Hong-Jie Dong

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

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 solutio...