Kato's inequality and Kato's inequality up to the boundary (Q950121)

Let \(U\) be a domain in \(\mathbb{R}^{d}\). The classical form of \textit{T. Kato}'s inequality [Isr. J. Math. 13(1972), Proc. internat. Sympos. partial diff. Equ. Geometry normed lin. Spaces I, 135--148 (1973; Zbl 0246.35025)] says that, if \(u\) and \(\Delta u\) belong to \(L^{1}(U)\), then \(\Delta u^{+}\) is a measure and \( \Delta u^{+}\geq \mathbf{1}_{\{u\geq 0\}}\Delta u\). This has been generalized by several authors. Recently, \textit{H. Brezis} and \textit{A. C. Ponce} [Commun. Contemp. Math. 10, 1217--1241 (2008; Zbl 1162.31005)] have shown that, if \(u\in W^{1,1}(U)\), where \(U\) is bounded and smooth, and if \(\Delta u\) is a finite measure in \(U\) and the weak normal derivative \(\partial _{n}u\) is a measure on \(\partial U\), then \(\Delta u^{+}\) and \(\partial _{n}u^{+}\) are finite measures and \( \left\| \Delta u^{+}\right\| +\left\| \partial _{n}u^{+}\right\| \leq \left\| \Delta u\right\| +\left\| \partial _{n}u\right\| \). The author answers a question raised in the latter paper by showing that, under these hypotheses, \(\partial _{n}u^{+}=\mathbf{1}_{\{u>0\}}\partial _{n}u-\mathbf{1}_{\{u=0\}}(\partial _{n}u)^{-}\) when \(\{u>0\}\) and \(\{u=0\}\) are suitably defined. The proofs are potential theoretic in nature and the results are established for a class of symmetric second order elliptic operators in divergence form.