## 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.

### MSC:

 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
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