Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga-type functionals

Motivated by the formula, due to Bourgain, Brezis and Mironescu, (equation presented) that characterizes the functions in L^q that belong to W^1,q (for q > 1) and BV (for q = 1), respectively, we study what happens when one replaces the denominator in the expression above by |x - y|. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV function. We further identify the function space relevant to the study of these functionals, the space BVq, as the Besov space B_q,∞ ^1/q. We show, among other things, that BVq(Ω) contains both the spaces BV (Ω) ∩ L^∞(O) and W^1/q,q(Ω). We also present applications to the study of singular perturbation problems of Aviles-Giga type.