Partial regularity result of elliptic systems with Dini continuous coefficients and $q$-growth

TL;DR

This paper proves partial regularity for vector solutions to second order elliptic systems with Dini continuous coefficients and super-quadratic growth, showing solutions are smooth outside a singular set.

Contribution

It establishes a new partial regularity result for elliptic systems with Dini continuous coefficients and q-growth, extending previous regularity theories.

Findings

Solutions are $C^1$ outside a singular set.

Regularity holds under Dini continuity with super-quadratic growth.

The singular set has measure zero or is small in a specific sense.

Abstract

We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of the solutions outside of singular set.