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Zbl 0857.35126
Boccardo, Lucio; Gallouët, Thierry; Orsina, Luigi
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, No.5, 539-551 (1996). ISSN 0294-1449

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 $(*)$.

MSC 2000:: *35R05 PDE with discontinuous coefficients or data
47H05 Monotone operators (with respect to duality)
35J60 Nonlinear elliptic equations

Keywords: $p$-capacity; decomposition theorem; existence and uniqueness; entropy solutions

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