Coarea formulae and chain rules for the Jacobian determinant in\n fractional Sobolev spaces

2019/03/18 by Peter Gladbach, Gladbach, Peter, Heiner Olbermann +1
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1903.07420

Abstract

We prove weak and strong versions of the coarea formula and the chain rule\nfor distributional Jacobian determinants Ju for functions u in fractional\nSobolev spaces W^s,p(\Ω), where \Ω is a bounded domain in\n\ℝⁿ with smooth boundary. The weak forms of the formulae are proved\nfor the range sp>n-1, s> \(n-1)/(n), while the strong versions are\nproved for the range sp\≥ n, s\≥ \(n)/(n+1). We also provide a chain\nrule for distributional Jacobian determinants of H "older functions and point\nout its relation to two open problems in geometric analysis.\n

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