Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with discontinuous data

TL;DR

This paper establishes optimal Morrey regularity for weak solutions to certain quasilinear elliptic systems with irregular boundary data, advancing understanding of solution regularity under minimal boundary smoothness.

Contribution

It introduces a method to prove boundedness and optimal Morrey regularity of solutions for quasilinear systems with discontinuous data and irregular boundaries.

Findings

Weak solutions are bounded under given conditions.

Optimal Morrey regularity of the gradient is achieved.

Applicable to systems with irregular boundary data.

Abstract

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carath\'eodory maps having Morrey regularity in $x$ and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient.