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Nonlinear elliptic and parabolic equations involving measure data. (English) Zbl 0707.35060

Let ${\Omega }$ be a nonempty bounded set in ${ℝ}^{N}$. The authors prove the existence of solutions to

$\left(E\right)\phantom{\rule{1.em}{0ex}}Au=f\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },$

where $Au=-div\left(a\left(x,Du\right)\right)$ with a: ${\Omega }×{ℝ}^{N}\to {ℝ}^{N}$ 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 ${W}_{0}^{1,p}\left({\Omega }\right)$ for f in ${W}^{-1},{p}^{\text{'}}\left({\Omega }\right)$ and then obtaining estimates on u which depend only on ${\Omega }$, a and ${\parallel f\parallel }_{{L}^{1}}$. Finally, f is approximated by a sequence in ${W}^{-1/{p}^{\text{'}}}\left({\Omega }\right)$. Extension to the equation

$Au+g\left(x,u\right)=f\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}u=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }$

and a parabolic analog of (E) is also given.

Reviewer: P.K.Wong

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial value problems for linear parabolic equations 35Dxx Generalized solutions of PDE 35B45 A priori estimates for solutions of PDE
##### Keywords:
measure data; quasilinear