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Fractional Sobolev regularity for the Brouwer degree


De Lellis, Camillo; Inauen, Dominik (2017). Fractional Sobolev regularity for the Brouwer degree. Communications in Partial Differential Equations, 42(10):1510-1523.

Abstract

We prove that if Ω⊂ℝn is a bounded open set and nα>dimb(∂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,⋅)∉Wβ,p, where is the unit ball.

Abstract

We prove that if Ω⊂ℝn is a bounded open set and nα>dimb(∂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,⋅)∉Wβ,p, where is the unit ball.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:31 October 2017
Deposited On:18 Jan 2018 11:06
Last Modified:18 Apr 2018 11:49
Publisher:Taylor & Francis
ISSN:0360-5302
Funders:Schweizerischer Nationalfonds
Additional Information:This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Partial Differential Equations on 06 Oct 2017, available online: http://wwww.tandfonline.com/10.1080/03605302.2017.1380040.
OA Status:Green
Publisher DOI:https://doi.org/10.1080/03605302.2017.1380040

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