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Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. (English) Zbl 0857.35126

Summary: We consider the differential problem

$A\left(u\right)=\mu \phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{2.em}{0ex}}u=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{2.em}{0ex}}\left(*\right)$

where ${\Omega }$ is a bounded, open subset of ${ℝ}^{N}$, $N\ge 2$, $A$ is a monotone operator acting on ${W}_{0}^{1,p}\left({\Omega }\right)$, $p>1$, and $\mu$ is a Radon measure on ${\Omega }$ that does not charge the sets of zero $p$-capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in ${L}^{1}\left({\Omega }\right)$ and of a measure in ${W}^{-1,{p}^{\text{'}}}\left({\Omega }\right)$), and an existence and uniqueness result for the so-called entropy solutions of $\left(*\right)$.

##### MSC:
 35R05 PDEs with discontinuous coefficients or data 47H05 Monotone operators (with respect to duality) and generalizations 35J60 Nonlinear elliptic equations