Scalar De Giorgi theory does not transfer verbatim to systems

The De Giorgi--Nash--Moser estimates on this page concern a single real-valued unknown. In De Giorgi-Nash interior Holder regularity for divergence-form equations, $u\in H^1(\mathrm{\Omega };\mathbb{R})$ solves the scalar equation $-\mathrm{div}(A\mathrm{\nabla }u)=0$, where $A$ is a measurable symmetric uniformly elliptic spatial matrix. These are the hypotheses in [V] §1, Theorem 1. A component of a coupled elliptic system need not satisfy this scalar equation, so the theorem cannot be applied to that component merely because the system has an ellipticity condition. In particular, a system condition such as Legendre--Hadamard ellipticity does not by itself check the scalar hypotheses of this page. Any application to components must separately verify those hypotheses, as one can for a decoupled collection of scalar equations. Regularity theory for coupled systems is outside this page's scope.

Velichkov, Elliptic PDEs: Teorema di De Giorgi, Section 1 and Theorem 1 (printed p. 1 of the complete 7-page note, read in full), states and proves the interior regularity theorem for a scalar real-valued solution $u$ of $-\mathrm{div}(A\mathrm{\nabla }u)=0$ with a symmetric uniformly elliptic matrix $A$; the statement has no vector-valued or system analogue. This item records only the resulting limitation of De Giorgi-Nash interior Holder regularity for divergence-form equations and asserts no system counterexample and no system regularity theorem.