Weighted Poincaré and Korn inequalities for Hölder α domains

Abstract. It is known that the classic Korn inequality is not valid for Hölder α domains. In this paper we prove a family of weaker inequalities for this kind of domains, replacing the standard Lp-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 Poincare ́ inequalities and then, generalizing an argument of Kondratiev and Oleinik, we show that weighted Korn inequalities can be derived from them. The Poincare ́ type inequalities proved here improve previously known results. We show by means of examples that our results are optimal. 1