Green’s theorem from the viewpoint of applications. (English) Zbl 1060.35504

The paper contains a new detailed proof of Green’s theorem for functions from the Sobolev space \(W^{1,p}\), \(1\leq p < \infty \), defined on bounded two-dimensional domains with a Lipschitz continuous boundary. A special attention is paid to internal an external cusp-points. Line integrals are defined in a natural way without any use of partition of the unity. Divergence forms of Green’s theorem are proved in detail as well.


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
35J20 Variational methods for second-order elliptic equations
65N99 Numerical methods for partial differential equations, boundary value problems
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