Maria Karmanova (Novosibirsk State University)
Area & coarea formulas on
rectifiable sets in metric spaces and Sobolev mappings
Abstract.
We study the metric differentiability of mappings
defined on measurable subsets of $\mathbb R^n$ with values in an
arbitrary complete metric space. Using results for Lipschitz mappings we
investigate the metric differentiability of more general mappings
including Sobolev ones. We prove also Stepanov type theorem on
metric differentiability of mappings. We give suitable definitions
of coarea factor both for mappings defined on
$n$-rectifiable metric spaces with the
values in a $k$-rectifiable metric space, $k\leq n$,
and of Jacobian
of mappings defined on $n$-rectifiable metric space
with the range in an arbitrary complete metric space.
We give a direct proof of the
area and coarea formulas for mappings under consideration without
an isometric embedding in a Banach space.
We generalize the area and
coarea formulas for Sobolev mappings defined on Riemannian
manifolds with values in complete metric spaces.
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