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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