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

Abstract

We develop an abstract regularity framework for a class of two-dimensional nonlinear elliptic systems, including almost harmonic maps. The approach combines a Campanato-type iteration scheme with a Caccioppoli-type estimate and identifies general assumptions under which local H{\"o}lder continuity follows. More precisely, we prove that any class of admissible pairs $(u,f)$ that is stable under rescaling and satisfies an oscillation-decay property consists of locally H{\"o}lder continuous maps. The resulting H{\"o}lder exponent is explicit and matches the classical Morrey--Campanato threshold determined by the Lebesgue integrability of the source term $f$. The framework is purely analytic and avoids the use of $\mathcal{H}^1$--$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization. We illustrate the flexibility of the framework through several classes of elliptic systems. As a first example, we recover local H{\"o}lder continuity for almost harmonic maps \[ -\Delta u=|\nabla u|^2u+f \] into $\mathbb{S}^n$ with $L^q$-integrable tension fields by means of a direct argument independent of the classical harmonic map regularity theory. We next consider systems of the form \[ -\Delta u=\Omega\cdot\nabla u+f, \] showing that the analytic condition $\operatorname{div}\Omega\in L^q$ for some $q>1$ is sufficient to ensure regularity (for classical harmonic maps, $\operatorname{div}\Omega=0$). We further apply the framework to...