$\mathcal{A}$-harmonic approximation and partial regularity, revisited

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their ${\mathrm{C}}^{1,\alpha }$ or ${\mathrm{C}}^{\mathrm{\infty }}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the ${\mathrm{L}}^p$-theory for strongly elliptic linear systems and Sobolev’s embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.