Interpolation inequalities for partial regularity

Abstract We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p > 1 {p>1} . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.