Nonlinear measure data problems
This paper is a survey of some regularity results for solutions to measure data problems, i.~e. solutions of the equation \(-\text{div}\,a(x,Du) = \mu\), where \(\mu\) is a measure on a bounded open set \(\Omega \subset \mathbb{R}^{n}\). After two introductory sections, the basic regularity results are given. These results are new in the sublinear case, \(p<2\) (the case \(p\geq2\) was treated by the author in [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 2, 195--261 (2007; Zbl 1178.35168)]), and they are proved in the last sections of the paper. Next, based on the concept of Wolff potential, nonlinear extensions of the basic linear estimates via the Riesz potential are presented. Also presented are recently obtained nonlinear potential estimates for parabolic problems \(u_{t}-\text{div}(a(Du))=\mu\).
scientific article; zbMATH DE number 5990865
scientific article