The co-area formula for Sobolev mappings

Abstract

Abstract. We extend Federer's co-area formula to mappings f belonging to the Sobolev class W 1,p (R n ; R m ), 1 ≤ m < n, p > m, and more generally, to mappings with gradient in the Lorentz space L m,1 (R n ). This is accomplished by showing that the graph of f in R n+m is a Hausdorff n-rectifiable set.

The flux problem for the Navier-Stokes equations

“…The corresponding proof is based on the co-area formula (see, for instance, [50]). From Lemmas 3.5 and 5.2 we obtain the following result.…”

A note on Lusin's condition (N) for W_loc^1,n-mappings with convex potentials

“…The literature on Lusin's condition (N) is vast and we will only mention a few essential results to establish some context. For instance, in [10, Corollary B], Malý and Martio proved that C(Ω) ∩ W 1,n loc (Ω) ∩ {F : Ω → R n : F open} ⊂ N(Ω) and, in [10,Theorem C], that C α loc (Ω) ∩ W 1,n loc (Ω) ⊂ N(Ω) for every α ∈ (0, 1) (see also Malý's Theorem 1.3 in [9]). However, C(Ω) ∩ W 1,n loc (Ω) ⊂ N(Ω) (see [10, Section 1] and references therein).…”

Задача Протекания Для Уравнений Навье - Стокса

“…Справедлива следующая лемма. Доказательство этой леммы основано на применении формулы коплощади (см., например, [50]).…”