The Calderón-Zygmund theory for elliptic problems with measure data. (English) Zbl 1178.35168

Mingione, Giuseppe

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 2, 195-261 (2007).

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.

Reviewer: Lubomira Softova (Aversa)

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

Keywords:

non-linear elliptic equations; Radon measure; Dirichlet problem; Marcinkiewicz spaces; Morrey spaces; VMO