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

Let Ω be a nonempty bounded set in N . The authors prove the existence of solutions to

(E)Au=finΩ,u=0onΩ,

where Au=-div(a(x,Du)) with a: Ω× N 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 (Ω) for f in W -1 ,p ' (Ω) and then obtaining estimates on u which depend only on Ω, a and f L 1 . Finally, f is approximated by a sequence in W -1/p ' (Ω). Extension to the equation

Au+g(x,u)=finΩ,u=0onΩ

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

Reviewer: P.K.Wong

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35K60Nonlinear initial value problems for linear parabolic equations
35DxxGeneralized solutions of PDE
35B45A priori estimates for solutions of PDE