Measure data systems with Orlicz growth

TL;DR

This paper investigates the existence and regularity of very weak solutions to nonlinear elliptic systems with measure data and Orlicz growth, extending results beyond superquadratic cases and providing conditions for Sobolev regularity.

Contribution

It establishes existence, regularity estimates, and Sobolev criteria for solutions to systems with Orlicz growth and measure data, broadening the scope beyond superquadratic growth.

Findings

Solutions exist under Orlicz growth conditions.

Regularity estimates are provided in the Marcinkiewicz scale.

A precise condition ensures solutions are Sobolev functions.

Abstract

We study the existence of very weak solutions to a system \[\begin{cases}-\mathrm{div} \mathcal{A}(x,D\mathbf{u})=\mathbf{\mu}\quad\text{in }\ \Omega, \mathbf{u}=0\quad\text{on }\ \partial\Omega\end{cases} \] with a datum $\mathbf{\mu}$ being a vector-valued bounded Radon measure and $\mathcal{A}$ having measurable dependence on the spacial variable and Orlicz growth with respect to the second variable. We are {\em not} restricted to the superquadratic case. For the solutions and their gradients we provide regularity estimates in the generalized Marcinkiewicz scale. In addition, we show a precise sufficient condition for the solution to be a~Sobolev function.