Regularity of elliptic systems with Uhlenbeck structure

arXiv:2611.08635v1 Announce Type: new Abstract: We study autonomous elliptic systems with Uhlenbeck structure of the form \[ -\operatorname{div}\big(\mathfrak a(|Du|)Du\big)=0 \quad \text{in }\ \Omega\subset \mathbb R^n \] Under two-sided quantitative $(p,q)$-growth conditions on $\mathfrak a$, we prove gradient H\"older continuity with constants and exponent depending only on the dimensions, $p$ and $q$. This improves the classical theory initiated by Uhlenbeck by removing the continuity requirement on $\mathfrak a'$. The proof uses nonlinear comparison systems whose regularity follows from a scalar uniformly elliptic equation. Under weaker $(p,q)$-growth conditions on the convex function $\varphi(t):=\int_0^t s\mathfrak a(s)\,ds$, we obtain local $W^{1,s}$-regularity for every $s>1$, and hence H\"older continuity of solutions with every exponent less than one. An additional lower growth condition on $\mathfrak a$ yields local Lipschitz continuity, even when $\mathfrak a$ is discontinuous.

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