Preprint

Inserted: 21 jun 2026
Last Updated: 21 jun 2026

Pages: 59
Year: 2026

Abstract:

We establish interior regularity results for a broad class of two-dimensional nonlinear elliptic systems. Our approach isolates the core integrability mechanism within a unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate. Specifically, we show that within any class of admissible pairs $(\boldsymbol{u}, \boldsymbol{f})$ that is stable under rescaling and satisfies a discrete oscillation-decay axiom, the map $\boldsymbol{u}$ is automatically locally H\"older continuous. Furthermore, the resulting H\"older exponent is explicit and optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term $\boldsymbol{f}$. This purely analytic framework systematically avoids the $\mathcal{H}^1$--$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization techniques that underpin existing regularity theories. We apply this principle to derive regularity results in regimes lying strictly beyond the reach of existing gauge-theoretic methods. As a foundational example, we provide a new direct proof of local H\"older continuity for almost harmonic maps $-\Delta \boldsymbol{u}= | \nabla \boldsymbol{u} |^2 \boldsymbol{u} + \boldsymbol{f}$ into $\mathbb{S}^n$ with $L^q$-integrable tension fields. We then extend the analysis to systems of the form $-\Delta \boldsymbol{u} = \Omega \cdot \nabla \boldsymbol{u} + \boldsymbol{f}$, replacing geometric antisymmetry assumption on the connection form $\Omega \in L^2$ with the purely analytic condition $\mathrm{div} \Omega \in L^q$ for some $q>1$...

Keywords: Micromagnetics, Magnetic Skyrmions, Weakly harmonic maps, Interior regularity, Campanato iteration, Gauge transformations

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• Manuscript: A Classical Elliptic Regularity Approach to Almost Harmonic Maps and Related Systems