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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 $$A(u)=\mu\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega,\tag$*$ $$ where $\Omega$ is a bounded, open subset of $\bbfR^N$, $N\geq 2$, $A$ is a monotone operator acting on $W_0^{1,p}(\Omega)$, $p>1$, 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 $L^1(\Omega)$ and of a measure in $W^{-1,p'}(\Omega)$), and an existence and uniqueness result for the so-called entropy solutions of $(*)$.

35R05PDEs with discontinuous coefficients or data
47H05Monotone operators (with respect to duality) and generalizations
35J60Nonlinear elliptic equations
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