A counterexample to Schauder estimates for elliptic operators with unbounded coefficients

Priola, Enrico

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 12 (2001) no. 1, pp. 15-25

Résumé

We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space $\mathbb{R}^{2}_{+}$ of $\mathbb{R}^{2}$. We show that for a particular initial datum, which is Lipschitz continuous and bounded on $\mathbb{R}^{2}_{+}$, the second derivative of the classical solution is not uniformly continuous on $\mathbb{R}^{2}_{+}$. In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.