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

Let \(\Omega\) be a nonempty bounded set in \({\mathbb{R}}^ N\). 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: \(\Omega\times {\mathbb{R}}^ N\to {\mathbb{R}}^ N\) 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 \(W_ 0^{1,p}(\Omega)\) for f in \(W^{-1},p'(\Omega)\) and then obtaining estimates on u which depend only on \(\Omega\), a and \(\| f\|_{L^ 1}\). Finally, f is approximated by a sequence in \(W^{-1/p'}(\Omega)\). 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.
Reviewer: P.K.Wong


35J65 Nonlinear boundary value problems for linear elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Dxx Generalized solutions to partial differential equations
35B45 A priori estimates in context of PDEs
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