Fractional Sobolev regularity for the Brouwer degree

Camillo De Lellis, Dominik Inauen
2017 Communications in Partial Differential Equations  
We prove that if Ωn is a bounded open set and n>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 0 and p1 with there is a vector field with deg (v,Ω,)W,p, where is the unit ball. Abstract. We prove that if Ω ⊂ R n is a bounded
more » ... set and nα > dim b (∂Ω) = d, then the Brouwer degree deg(v, Ω, ·) of any Hölder function v ∈ C 0,α (Ω, R n ) belongs to the Sobolev space W β,p (R n ) for every 0 ≤ β < n p − d α . 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 β > n p − n−1 α there is a vector field v ∈ C 0,α (B1, R n ) with deg (v, Ω, ·) / ∈ W β,p , where B1 ⊂ R n is the unit ball.
doi:10.1080/03605302.2017.1380040 fatcat:uifgtoqisjhnnns2mmxrp4hq4a