Recent results on quasilinear differential equations. I. (Q2846676)

This paper concerns regularity of the weak solutions of the \(p\)-Laplacian with all standard boundary conditions (Dirichlet, Neumann and Robin) on possibly non-smooth domains. The paper discusses the interplay of the boundary conditions and the smoothness of the domain to get the desired regularity. It is a survey paper of joint results in [\textit{D. Daners} and \textit{P. Drábek}, Trans. Am. Math. Soc. 361, No. 12, 6475--6500 (2009; Zbl 1181.35098)]. To make the exposition clear, the author focuses only on the \(L^{\infty }\)-estimates for weak solutions for the \(p\)-Laplacian. \(C^{1,\alpha }\)-regularity and maximum principle for the weak solutions are presented as an application. Finally, existence, continuity and compactness of the resolvent operator is proved.NEWLINENEWLINEFor the entire collection see [Zbl 1262.35002].