Measure data systems with Orlicz growth

Abstract

We study the existence of very weak solutions to a system \[\begin{cases}-\mathrm{div} \mathcal{A}(x,D\mathbf{u})=\mathbfμ\quad\text{in }\ Ω, \mathbf{u}=0\quad\text{on }\ \partialΩ\end{cases} \] with a datum $\hat{\mathbf{I}}¼$ 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.

We supply potential estimates arxiv:2102.09313 with existence for very weak solutions to broader class of systems