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

Abstract

Motivated by the formula, due to Bourgain, Brezis and Mironescu,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 BV q , as the Besov space B 1/q q,∞ . We show, among other things, that BV q (Ω) contains both the spaces BV (Ω) ∩ L ∞ (Ω) and W 1/q,q (Ω). We also present applications to the study of singular perturbation problems of Aviles-Giga type.

Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung

“…The following technical proposition makes the connection between Besov spaces and the quantities appearing in the statement of Theorem 1.5. This proposition is a direct consequence of Corollary A.3 and Lemma A.5, see in the Appendix, whose proofs are based on similar arguments to those used in [5].…”

“…The next Proposition is proved exactly as a similar statement in [5]; the proof is postponed to the Appendix; in both cases the key ingredient is Proposition A.1 which is part of [5, Proposition…”

Asymptotic behavior of theW1∕q,q-norm of mollifiedBVfunctions and applications to singular perturbation problems

“…as in (1.2) appears, was obtained in [13]. More precisely, we showed in [13] that for every radial η ∈ C ∞ c (R N , R) there exists a constant C = C η > 0 such that for every u ∈ BV (Ω, R d ) ∩ L ∞ we have lim ε→0 + ε η ε * u More recently, we showed in [14] yet another related result: with the dimensional constant C N > 0 defined by…”

Optimal Besov Differentiability for Entropy Solutions of the Eikonal Equation

“…3,∞ are limits of Aviles-Giga sequences, see [35]). Moreover our kinetic formulation (see (KIN) below) takes a simpler form than (1.2).…”