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, \begin{equation*} \lim_{\varepsilon\to 0^+} \int_Ω\int_Ω\frac{|u(x)-u(y)|^q}{|x-y|^q}\,ρ_\varepsilon(x-y)\,dx\,dy=K_{q,N}\|\nabla u\|_{L^{q}}^q\,, \end{equation*} 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 $B{V}^q$, as the Besov space $B_{q,\infty }^{1/q}$. We show, among other things, that $B{V}^q(\Omega )$ contains both the spaces $BV(\Omega )\cap L^{\infty }(\Omega )$ and $W^{1/q,q}(\Omega )$. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

Accepted in Communications in Contemporary Mathematics

References in corpus (1)

• On formulae decoupling the total variation of BV functions

Cited by in corpus (1)

• Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions