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Extra resources for Amorphous Semiconductors, 2nd Edition (Topics in applied physics, vol. 36)

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25) because the mean values UBi 1 = udJ1- TBi converge to u(x) as i ~ 00. In Eq. 25), so the sum on the right-hand side is at least t = C(€)t L a- iE ~ C(€)tr i~O E L r: i~O for some positive constant C(€). ;. ; this can be done by the properties of the chain. (Note that BI. 26), and by invoking the basic covering theorem. 18. 4. 27. 11), at least when the true Sobolev conjugate np/(n - p) is replaced by any number r < np/(n - p). 18. 28 Notes to Chapter 4. The approach to Poincare inequalities by way of thick curve families has been emphasized by David and Semmes [38], [160]; the method can be used very generally.

39 Notes to Chapter 3. Most of the material in this chapter is standard and can be found in many books. Some of them are listed in the bibliography. I recommend [45] and [205] as friendly geometric introductions to the subject; these books also contain basics of the theory of B V functions. Extensive treatments of Sobolev spaces are the monographs by Maz'ya [136] and by Adams and Hedberg [I]. Chapter 6 in [198] contains a nice elementary discussion of differential forms and Hodge theory. For the nonsmooth LP theory, the classical reference is [142].

3) vanish on the bails Bi and B;. 24. 23 exists. 25 Notes to Chapter 5. This chapter is taken for the most part from the two papers [71], [72] of Hajlasz. The theory of Hajlasz-Sobolev spaces is currently taking shape, and it would be unreasonable to go deeper into this study here. Papers on the topic include [73], [75], [76], [106], [108], [113], and [167]. 6 Lipschitz Functions Lipschitz functions are the smooth functions of metric spaces. 1) for all x and y in X. 1) holds. It is important to observe that in every metric space there are plenty of nontrivial real-valued Lipschitz functions (unless the space itself is somehow trivial such as a point).

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