Regularity of elliptic systems with Uhlenbeck structure

Abstract

We study autonomous elliptic systems with Uhlenbeck structure of the form $-\operatorname{div}\big(\mathfrak a(|Du|)Du\big)=0 \quad \text{in }\ Ω\subset \mathbb R^n$ Under two-sided quantitative $(p,q)$-growth conditions on $\mathfrak a$, we prove gradient Hölder 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 $φ(t):=\int_0^t s\mathfrak a(s)\,ds$, we obtain local $W^{1,s}$-regularity for every $s>1$, and hence Hölder 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.