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The Calderón-Zygmund theory for elliptic problems with measure data. (English) Zbl 1178.35168
The author considers the following Dirichlet problem in a bounded domain:
\[ -\text{ div}\, a(x,Du)=\mu \;\text{ in} \;\Omega,\quad u=0 \;\text{ on} \;\partial\Omega\,, \]
where \(\mu\) is a signed Radon measure with finite total variation \(|\mu|(\Omega)<\infty\) and \(a\) is a Carathéodory vector field. There are obtained differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are given and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build suitable Calderòn-Zygmund theory for the problem. All the regularity results are provided together with explicit local a priori estimates.

35J61 Semilinear elliptic equations
35J70 Degenerate elliptic equations
35R06 PDEs with measure
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
Full Text: arXiv