Titlebook: Weakly Differentiable Functions; Sobolev Spaces and F William P. Ziemer Textbook 1989 Springer Science+Business Media New York 1989 Derivat

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Preliminaries, information contained in this chapter will be well-known by the reader and therefore no attempt has been made to make a complete and thorough presentation. Rather, we merely introduce notation and develop a few concepts that will be needed in the sequel.

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Sobolev Spaces and Their Basic Properties,e basic Sobolev inequality is proved in two ways, one of which employs the co-area formula (Section 2.7) to obtain the best constant in the inequality. This method relates the Sobolev inequality to the isoperimetric inequality.

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Graduate Texts in Mathematicshttp://image.papertrans.cn/w/image/1021369.jpg

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,Poincaré Inequalities—A Unified Approach,In Chapter 2, basic Sobolev inequalities were established for functions in the space .(Ω). We recall the following fundamental result which is a particular case of Theorem 2.4.2.