Fractional Sobolev regularity for the Brouwer degree. Issue 10 (3rd October 2017)

• Record Type: Journal Article
• Authors: De Lellis, Camillo; Inauen, Dominik
• Is Part Of: Communications in partial differential equations. Volume 42:Issue 10(2017)
• Page Start: 1510
• Page End: 1523
• Publication Date: 2017-10-03
• Subjects: Brouwer degree -- fractional Sobolev regularity -- Hölder functions; 26B10 -- 55M25 -- 46E35; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2017.1380040
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We prove that if Ω⊂ℝ n is a bounded open set and nα > dim b ( ∂Ω ) = d, then the Brouwer degree deg( v, Ω, ⋅) of any Hölder function belongs to the Sobolev space for every . This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover, we show the optimality of the range of exponents in the following sense: for every β ≥0 and p ≥1 with there is a vector field with deg ( v, Ω, ⋅)∉ W β, p, where is the unit ball.