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

Abstract

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants Ju for functions u in fractional Sobolev spaces W s,p (Ω), where Ω is a bounded domain in R n with smooth boundary. The weak forms of the formulae are proved for the range sp > n − 1, s > n−1 n , while the strong versions are proved for the range sp ≥ n, s ≥ n n+1 . We also provide a chain rule for distributional Jacobian determinants of Hölder functions and point out its relation to two open problems in geometric analysis.

An estimate of the Hopf degree of fractional Sobolev mappings

“…Proof. Our proof is based on trace estimates for harmonic extensions, see [7], [22, (2.3)], [19, Section 10], [9]. Alternatively, one could resort to heavier Harmonic Analysis (namely Littlewood-Paley projections and paraproducts), as in [26], or to Fourier Analysis, as in [28], to obtain the same estimate.…”

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

“…To put our results in their proper context, our results must be compared with the statements in [29,22,30,1,2,31,6,7,16,9,5,10,25] on isometric immersions, in [19,28,35,23] on the rigidity of Sobolev solutions to the Monge-Ampère equation, and in [3,20,21,13,26,25] on geometric or topological properties of weakly regular deformations. 1.2.…”

“…The question remains open for the range 1/5 ≤ α ≤ 2/3. Gromov [14, Section 3.5.5.C, Open conjectures the critical exponent to be α = 1/2, see also [13].…”

Global Nash-Kuiper theorem for compact manifolds

“…A short modern proof based on regularization and a commutator estimate was provided in [12]. More recently there has been intensive work on lowering the rigidity exponent [16,23], one conjecture being that some form of rigidity should hold for all θ > 1/2 [16,21,26].…”