By Yu. G. Reshetnyak (auth.)
This is often one of many first monographs to accommodate the metric concept of spatial mappings and contains leads to the speculation of quasi-conformal, quasi-isometric and different mappings.
the most topic is the learn of the steadiness challenge in Liouville's theorem on conformal mappings in area, that's consultant of a few difficulties on balance for transformation periods. To permit this research quite a lot of mathematical instruments has been constructed which comprise the calculus of version, estimates for differential operators like Korn inequalities, houses of capabilities with bounded suggest oscillation, and so forth.
effects received through others learning related subject matters are pointed out, and a survey is given of guides treating appropriate questions or related to the approach proposed.
This quantity might be of significant worth to graduate scholars and researchers attracted to geometric functionality conception.
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This is often one of many first monographs to house the metric conception of spatial mappings and contains leads to the speculation of quasi-conformal, quasi-isometric and different mappings. the most topic is the learn of the soundness challenge in Liouville's theorem on conformal mappings in house, that's consultant of a few difficulties on balance for transformation periods.
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Additional info for Stability Theorems in Geometry and Analysis
The differential f' (x) is the linear mapping for Xn < 0 and the mapping > O. ) = 1 for Xn > O. 1) for almost all x E Rn with the constant K = 1 and f belongs to the class W~,loc(Rn). ) for its Jacobian changes the sign. The class of mappings that appears if, in the definition of a mapping with bounded distortion, we remove the condition for the Jacobian to have constant sign in the domain of definition of the mapping can contain mappings with however involved singularities and, apparently, cannot be the subject of a substantive theory.
One more difficulty arising in the proof of the lemma relates to the fact that we should also take into account the requirement that the value PC,Q(g - 1) vanishes.
The notion of space quasi conformal mapping was introduced by M. A. Lavrent 'ev as long ago as 1938 (see [69)). However, the beginning of intensive investigations in this area dates from 1960. Among many articles devoted to the topic we should distinguish the fundamental research by F. Gehring . A survey of results on space quasi conformal mappings which were accomplished until 1972 is given, for instance, in the monograph  (see also [29)). The original definition of space quasi conformal mappings was purely geometric.