Sobolev and isoperimetric inequalities with monomial weights

详细信息    查看全文

• 作者：Xavier Cabr¨¦ ; Xavier Ros-Oton
• 关键词：Weighted Sobolev inequality ; Isoperimetric inequalities with a density ; Monomial weight ; Axial symmetries
• 刊名：Journal of Differential Equations
• 出版年：2013
• 出版时间：1 December, 2013
• 年：2013
• 卷：255
• 期：11
• 页码：4312-4336
• 全文大小：373 K

文摘

We consider the monomial weight in , where is a real number for each , and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by , and they contain the best or critical exponent (which depends on ). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions.

When are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space (with no weight) when written for axially symmetric functions and domains in .