We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space of R^2. We show that for a particular initial datum, which is Lipschitz continuous and bounded, the second derivative of the classical solution is not uniformly continuous. In particular this implies that the well known maximal Holder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.
A counterexample to Schauder estimates for elliptic operators with unbounded coefficients
PRIOLA, Enrico
2001-01-01
Abstract
We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space of R^2. We show that for a particular initial datum, which is Lipschitz continuous and bounded, the second derivative of the classical solution is not uniformly continuous. In particular this implies that the well known maximal Holder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.
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