By Heinonen J.
We survey fresh advances in research and geometry, the place first order differential research has been prolonged past its classical soft settings. Such experiences have functions to geometric stress questions, yet also are of intrinsic curiosity. The transition from soft areas to singular areas the place calculus is feasible parallels the classical improvement from delicate capabilities to features with vulnerable or generalized derivatives. additionally, there's a new manner of taking a look at the classical geometric conception of Sobolev features that's invaluable in additional normal contexts.
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We survey contemporary advances in research and geometry, the place first order differential research has been prolonged past its classical tender settings. Such reviews have purposes to geometric tension questions, yet also are of intrinsic curiosity. The transition from delicate areas to singular areas the place calculus is feasible parallels the classical improvement from tender services to capabilities with vulnerable or generalized derivatives.
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Additional resources for Nonsmooth calculus
4. As defined, ρu depends a priori on the fixed parameter p ∈ [1, ∞), although this dependence is supressed in the notation. In many situations, although not always, ρu is known to be independent of p (see Section 12). 5. Sobolev spaces N 1,p (X). Consider the vector space N consisting of all functions u : X → R such that u is in Lp (X) and there exists an upper gradient ρ of u in Lp (X). 6) ||u||1,p := ||u||Lp (X) + ||ρu ||Lp (X) , where ρu is the minimal p-weak upper gradient of u defined in the previous subsection.
NONSMOOTH CALCULUS 51 for every ball B in X, for every function u : X → R, and for every upper gradient ρ of u. 1) holds in X is to require that X has plenty of rectifiable curves, uniformly at all scales. The parameter p measures in a subtle way the amount of curves; it is akin to the parameter p in the definition of modulus. In fact, the validity of a Poincar´e inequality can sometimes be stated in terms of modulus. In general, the connection is more suggestive than formal. We have seen in Section 6 that Rn supports a p-Poincar´e inequality for each p ≥ 1.
Notes. Spivak’s book  contains a nice analysis of Riemann’s lecture. Of course, there is a huge literature on this topic. The most immediate generalization of a Riemannian metric leads to Finsler geometry, where one equips each tangent space of a smooth manifold with an arbitrary (smoothly varying) Banach norm; see, for example, . Basic references to Lipschitz manifolds are , , . The monographs ,  contain much information about spaces with bounded curvature in the sense of Alexandrov.