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, \begin{equation*} \lim_{\varepsilon\to 0^+} \int_\Omega\int_\Omega \frac{|u(x)-u(y)|^q}{|x-y|^q}\,\rho_\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 $BV^q$, as the Besov space $B^{1/q}_{q,\infty}$. We show, among other things, that $BV^q(\Omega)$ 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.