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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.

MSC:
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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References:
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