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Nonlinear elliptic and parabolic equations involving measure data. (English) Zbl 0707.35060
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.
Reviewer: P.K.Wong

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
Full Text: DOI
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