Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media

Abstract

We extend the concept of microlocal defect functionals to test functions belonging to the space $\mathrm{L}^\infty\cap\mathrm{VMO}_c$. Following L.~Tartar's remark that an extension of such concepts to $\mathrm{VMO}$ spaces should be possible, we establish a functional-analytic framework for this extension within the $\mathrm{L}^p-\mathrm{L}^q$ setting. Because the topological dual of $\mathrm{VMO}$ is the Hardy space $\mathcal{H}^1$, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict H\"older conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough $\mathrm{VMO}$ coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where $\sum_j a_j({\bf x})\xi_j=0$.