Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis) (Q837075)

Making use of potential theoretic methods, the interesting paper under review answers some questions raised by \textit{H. Brezis} and \textit{A. Ponce} [C. R. Math., Acad. Sci. Paris 338, No. 8, 599--604 (2004; Zbl 1101.35028), Commun. Contemp. Math. 10, No. 6, 1217--1241 (2008; Zbl 1162.31005)] which regard extensions of Kato's inequality and in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a \(W^{1,1}\)-function in a smooth domain. Precisely, the author studies the relations between the normal derivative of a function \(u\) and the normal derivative of its positive part \(u_{+}.\) The results apply to a large class of domains and elliptic operators in divergence form and, moreover, an expression of the normal derivative of a function of \(u\) is given. In an appendix, H.~Brezis solves an old conjecture by \textit{J. Serrin} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 385--387 (1964; Zbl 0142.37601)] about non-existence of pathological solutions of certain elliptic equations. With the aid of this result, the author succeeds to relax in a certain extent the required regularity assumptions in the paper's main results.