Estimates for the topological degree and related topics

Abstract

Abstract. This is a survey paper on estimates for the topological degree and related topics which range from the characterizations of Sobolev spaces and BV functions to the Jacobian determinant and nonlocal filter problems in Image Processing. These results are obtained jointly with Bourgain and Brezis. Several open questions are mentioned.Mathematics Subject Classification. 55M25, 49J99, 46E35, 46B50.

New characterizations of magnetic Sobolev spaces

“…This is motivated by the characterization of the Sobolev space H 1 (R N ) provided in [5,18] (see also [6][7][8][9][10][19][20][21][22]) in terms of the family of nonlocal functionals I δ which is defined by, for δ > 0,…”

Some characterizations of magnetic Sobolev spaces

“…where Q 2,N is the same positive constant appearing in (3) for p = 2. Other results related to the nonlocal operator I δ can be found in [9][10][11][12][13]. The aim of this note is to survey recent results contained in [14][15][16][17][18], where the authors have extended the aforementioned results to the magnetic setting.…”

Two subtle convex nonlocal approximations of the BV-norm

“…For works related to the BBM formula, see [5][6][7]15,16]. Other functionals converging to the BV-norm are considered in [3,8,9,[17][18][19][20][21][22].…”