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The Gauss-Green theorem in the context of Lebesgue integration. (English) Zbl 1089.26006

With proofs often motivated by the techniques employed in approaches using generalized Riemann integrals, the author, remaining entirely within the frame of Lebesgue integration, states and proves a general Gauss-Green theorem, which is essentially the type of Gauss-Green theorem obtained by the author using nonabsolute integration, with the supplementary assumption that the divergence of the vector field is Lebesgue integrable. Explicitly, the following result is proved.
Let \(A \subset {\mathbb R}^m\) be a bounded BV set, and let \(E_{0}\) and \(E_{\sigma}\) be slight and thin sets, respectively. Assume that a bounded vector field \(v : \text{cl}_{*}A \to {\mathbb R}^m\) is continuous in \(\text{cl}_{*}A \setminus E_{0}\) and pointwise Lipschitz in \(\text{cl}_{*}A \setminus E_{\sigma}\). If \(\text{div }v \in L^1(A),\) then \[ \int_{A} \text{div }v \,d{\mathcal L} = \int_{\partial_{*}A} v \cdot \nu_{A} \,d{\mathcal H}. \]
The essential closure \(\text{cl}_{*}A\) and essential boundary \(\partial_{*}A\) as well as the concepts of slight and thin sets are defined in the paper. Some applications are given.

MSC:

26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
26A45 Functions of bounded variation, generalizations
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
35J60 Nonlinear elliptic equations
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