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.

Abstract (VO)

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.

Riassunto

Si considera un problema ellittico di Dirichlet in un semispazio $\mathbb{R}^{2}_{+}$ di $\mathbb{R}^{2}$. In esso compare un operatore di tipo Ornstein-Uhlenbeck. Si dimostra, con calcoli espliciti, che per un particolare dato iniziale lipschitziano la corrispondente soluzione classica non ha la derivata seconda uniformemente continua su $\mathbb{R}^{2}_{+}$. Questo risultato implica in particolare che le ben note stime di Schauder non valgono in generale per problemi di Dirichlet su domini illimitati se i coefficienti sono illimitati.

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