arXiv subjects

Marin Mišur

Publications and source records attributed to Marin Mišur.

6 recordsLinked to original sources

arXiv2026

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

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 Hölder 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 $\sum_j a_j({\bf x})ξ_j=0$.

math.AP↗

arXiv2026

Sharp Volume-Based Bounds for BMO Functions on Compact Domains and Applications

We establish sharp localized $L^r$ bounds for compactly supported functions of bounded mean oscillation (BMO). Classical estimates obtained via the John-Nirenberg inequality bound the $L^r$ norm of such a function in terms of the diameter of its support. We show that the diameter dependence can be replaced by the Lebesgue measure of the support, the relevant quantity for thin, highly eccentric domains. The proof relies on non-increasing rearrangements and the Bennett-DeVore-Sharpley theorem, and requires no vanishing-oscillation hypothesis. We present physical and operator-theoretic applications: geometric collapse in tubular neighborhoods, a localized Cwikel-Lieb-Rozenblum bound for Schrödinger operators, and decoupling bounds for the localized action of Calderón-Zygmund singular integrals.

math.FA↗

arXiv2026

Global $L^p$ Second Commutation Lemma

We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1 < p < \infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available, the global $L^p$ setting has neither. We control the non-local tail through Calderón--Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-ε)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.

math.AP↗

arXiv2026

Logarithmic Resonance at Mixed Boundary Junctions in Gradient-Dependent Semilinear Equationsv

The regularity of solutions to elliptic partial differential equations degrades severely at mixed Dirichlet-Neumann boundary junctions, characterized classically by an $\mathcal{O}(r^{1/2})$ leading singular function. While this linear behavior is well documented, the introduction of gradient-dependent semilinear perturbations alters the local asymptotic profile. This article proves that gradient penalties scaling linearly with $|\nabla u|$ induce a highly localized $\mathcal{O}(r^{-1/2})$ source term that resonates with the half-integer spectrum of the principal homogeneous differential operator. This non-orthogonal resonance causes standard separable polynomial assumptions to fail at order $\mathcal{O}(r^{3/2})$. We establish a generalized resonance theorem that forces the emergence of a logarithmic anomaly, providing the exact analytical formulation of the resulting $r^{3/2} \ln(r)$ profile alongside rigorous local Sobolev regularity bounds for the remainder. By calculating the exact logarithmic coefficients and angular offsets for $\ell_1$, $\ell_2$, $\ell_\infty$, and arbitrary $\ell_q$-norm penalties, we demonstrate the universality of this obstruction. Finally, we formalize the corresponding enriched continuous Galerkin space (XFEM), establish its quasi-optimality, and present finite element experiments that confirm the predicted localized pollution, recover the predicted logarithmic coefficients across the $\ell_1$, $\ell_2$, and $\ell_\infty$ penalties on a curvature-free flat junction, and show that resolving the junction recovers optimal degree-of-freedom efficiency in standard finite element solvers; we close by outlining the targeted software architectures required for a fully enriched implementation.

math.NA↗

arXiv2026

On Phase-Space Orthogonality for Higher-Order Distributions: Obstructions and Algebraic Resolutions

The extension of microlocal defect measures to the $L^p$-$L^q$ framework necessitates the use of distributions which possess strictly positive finite order. A topological $\eps$-bounding definition of orthogonality for such distributions was recently introduced by Antonić, Mitrović, and Perić. Building on their framework, we systematically map the obstructions that arise when this natural approach is transported to strictly positive order, and we provide a complementary algebraic resolution. We first establish that while smooth spatial multipliers provide a robust framework for separating order-zero Radon measures, the partition-of-unity route to phase-space geometric separation breaks down for higher-order distributions: the separating frequency multiplier suffers an $\mathcal{O}(δ^{-κ})$ derivative divergence when frequency supports touch, which we prove defeats the a priori bound at every order $κ\ge 1$ and produces an explicit non-orthogonality already at order one. Transitioning to algebraic phase-space projectors reveals a second severe obstruction: it is mathematically impossible to separate scalar distributions algebraically, as one-dimensional smooth projectors enforce topological triviality. We demonstrate that the resolution is achieved exclusively within vector-valued PDE systems. Rather than relying on arbitrarily constructed, non-physical geometric scalar cut-offs $χ(x,ξ)$ to separate distinct modes, we redefine orthogonality canonically via zero-order algebraic matrix projectors. This replaces artificial geometric separation with the intrinsic spectral polarization of the governing system, yielding a coordinate-free microlocal orthogonality for systems of higher-order distributions.

math.AP↗

arXiv2022

Velocity averaging for diffusive transport equations with discontinuous flux

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to ``recognise'' changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for nonlinear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at $t=0$ for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.

math.AP↗