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

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 Hölder continuous. Furthermore, the resulting Hölder 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 Hölder continuity for almost harmonic maps $-Δ\boldsymbol{u} = |\nabla \boldsymbol{u}|^2 \boldsymbol{u} + \boldsymbol{f}$ into $\mathbb{S}^n$ with $L^q$-integrable tension fields. We then extend the analysis to systems of the form $-Δ\boldsymbol{u} = Ω\cdot \nabla \boldsymbol{u} + \boldsymbol{f}$, replacing geometric antisymmetry assumption on the connection form $Ω\in L^2$ with the purely analytic condition $\mathrm{div} Ω\in L^q$ for some $q>1$...