We prove that Omega subset of R-n is a bounded open set and n alpha > dim(b)(partial derivative Omega) = d then the Brouwer degree deg(v, Omega, .) of any Holder function v is an element of C-0,C-alpha (Omega, R-n) belongs to the Sobolev space W-beta,W-p (R-n) for every 0 <= beta < n/p - d/alpha. 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 beta >= 0 and p >= 1 with beta > n/p - n-1/alpha there is a vector field v is an element of C-0,C-alpha(B-1, R-n) with deg (v, Omega, .) is not an element of W-beta,W-p, where B-1 subset of R-n is the unit ball. ball.
Fractional Sobolev regularity for the Brouwer degree
De Lellis, C.;
2017-01-01
Abstract
We prove that Omega subset of R-n is a bounded open set and n alpha > dim(b)(partial derivative Omega) = d then the Brouwer degree deg(v, Omega, .) of any Holder function v is an element of C-0,C-alpha (Omega, R-n) belongs to the Sobolev space W-beta,W-p (R-n) for every 0 <= beta < n/p - d/alpha. 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 beta >= 0 and p >= 1 with beta > n/p - n-1/alpha there is a vector field v is an element of C-0,C-alpha(B-1, R-n) with deg (v, Omega, .) is not an element of W-beta,W-p, where B-1 subset of R-n is the unit ball. ball.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
