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Zbl 0707.35060
Boccardo, Lucio; Gallouët, Thierry
Nonlinear elliptic and parabolic equations involving measure data.
(English)
[J] J. Funct. Anal. 87, No.1, 149-169 (1989). ISSN 0022-1236

Let $\Omega$ be a nonempty bounded set in ${\bbfR}\sp N$. The authors prove the existence of solutions to $$(E)\quad Au=f\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega,$$ where $Au=-div(a(x,Du))$ with a: $\Omega\times {\bbfR}\sp N\to {\bbfR}\sp N$ is subject to certain coerciveness and monotonicity conditions and f is a bounded measure. This is done by first showing that (E) has a unique weak solution u in $W\sb 0\sp{1,p}(\Omega)$ for f in $W\sp{-1},p'(\Omega)$ and then obtaining estimates on u which depend only on $\Omega$, a and $\Vert f\Vert\sb{L\sp 1}$. Finally, f is approximated by a sequence in $W\sp{-1/p'}(\Omega)$. Extension to the equation $$Au+g(x,u)=f\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega$$ and a parabolic analog of (E) is also given.
[P.K.Wong]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35K60 (Nonlinear) BVP for (non)linear parabolic equations
35Dxx Generalized solutions of PDE
35B45 A priori estimates

Keywords: measure data; quasilinear

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