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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