A counterexample to Schauder estimates for elliptic operators with unbounded coefficients

Enrico Priola

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 1, page 15-25
  • ISSN: 1120-6330

Abstract

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We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space R + 2 of R 2 . We show that for a particular initial datum, which is Lipschitz continuous and bounded on R + 2 , the second derivative of the classical solution is not uniformly continuous on 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.

How to cite

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Priola, Enrico. "A counterexample to Schauder estimates for elliptic operators with unbounded coefficients." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.1 (2001): 15-25. <http://eudml.org/doc/252325>.

@article{Priola2001,
abstract = {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.},
author = {Priola, Enrico},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Optimal Hölder-regularity results; Dirichlet problems; The Ornstein-Uhlenbeck operator; Ornstein-Uhlenbeck operator},
language = {eng},
month = {3},
number = {1},
pages = {15-25},
publisher = {Accademia Nazionale dei Lincei},
title = {A counterexample to Schauder estimates for elliptic operators with unbounded coefficients},
url = {http://eudml.org/doc/252325},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Priola, Enrico
TI - A counterexample to Schauder estimates for elliptic operators with unbounded coefficients
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/3//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 1
SP - 15
EP - 25
AB - 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.
LA - eng
KW - Optimal Hölder-regularity results; Dirichlet problems; The Ornstein-Uhlenbeck operator; Ornstein-Uhlenbeck operator
UR - http://eudml.org/doc/252325
ER -

References

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