Weighted Poincaré and Korn inequalities for Hölder $\alpha$ domains

GABRIEL ACOSTA; RICARDO G. DURÁN; ARIEL L. LOMBARDI

MATHEMATICAL METHODS IN THE APPLIED SCIENCES

Wiley

Lugar: Chichester; Año: 2006 vol. 29 p. 387 - 400

0170-4214

It is known that the classic Korn inequality is not valid for Holder \alpha domains. In this paper, we prove a family of weaker inequalities for this kind of domains, replacing the standard L^p-norms by weighted norms where the weights are powers of the distance to the boundary. In order to obtain these results we prove first some weighted Poincaré inequalities and then, generalizing an argument of Kondratiev and Oleinik, we show that weighted Korn inequalities can be derived from them. The Poincaré type inequalities proved here improve previously known results. We show by means of examples that our results are optimal.