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Measure data systems with Orlicz growth

Iwona Chlebicka, Yeonghun Youn, Anna Zatorska-Goldstein

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 $\mathbfμ$ 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.

Measure data systems with Orlicz growth

Abstract

We study the existence of very weak solutions to a system with a datum being a vector-valued bounded Radon measure and 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.

Paper Structure

This paper contains 9 sections, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Basic definitions
  5. Notions of gradients and function spaces
  6. Embeddings
  7. The operator and definition of solutions
  8. Existence and Marcinkiewicz regularity to vectorial measure data problems
  9. Appendix

Theorems & Definitions (5)

  • proof
  • proof : Proof of Theorem \ref{['theo:exist']}
  • proof : Proof of Theorem \ref{['theo:SOLA']}
  • proof : Comments on the proof
  • proof