Boccardo, Lucio Elliptic and parabolic differential problems with measure data. (Italian) Zbl 0893.35131 Boll. Unione Mat. Ital., VII. Ser., A 11, No. 2, 439-461 (1997). This paper is a review of recent results obtained about elliptic and parabolic problems with measure data. Let \(\mu\) be a bounded measure and consider the problem \[ A(u) \equiv -\text{div} \bigl(a(x,u, \nabla u)\bigr) =\mu\quad\text{in }\Omega, \quad u=0 \text{ on } \partial \Omega, \tag{P} \] where \(\Omega\) is an open subset of \(\mathbb{R}^N\), \(a(x,s,\xi)\) is a Carathéodory function, monotone with respect to \(\xi\), \(A\) is a continuous and coercive operator of \(W_0^{1,2} (\Omega)\) into the dual space. If \(p\leq N\), solutions of (P) are not generally in \(W_0^{1,2} (\Omega)\) and may not be unique also when \(A\) is linear. In the linear case and \(p=2\) the problem has been studied by Stampacchia in the sixties. The nonlinear case has been recently investigated by many authors.The existence of weak solutions of (P) is proved by approximation. The uniqueness of these solutions is also proved. Moreover, a new and more general definition of solutions to problem (P), called entropy solutions, is introduced and uniqueness for such kind of solution is given, when \(\mu\) is a function in \(L^1\). An exhaustive bibliography of the relevant articles completes the paper. Reviewer: E.Mascolo (Firenze) Cited in 12 Documents MSC: 35R05 PDEs with low regular coefficients and/or low regular data 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations Keywords:approximation; uniqueness; entropy solutions PDFBibTeX XMLCite \textit{L. Boccardo}, Boll. Unione Mat. Ital., VII. Ser., A 11, No. 2, 439--461 (1997; Zbl 0893.35131)