## 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

### Keywords:

Green’s theorem; elliptic problems; variational problems
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### References:

 [1] G.M. Fichtengolc: Differential and Integral Calculus I. Gostechizdat, Moscow, 1951. [2] G.M. Fichtenholz: Differential- und Integralrechnung I. VEB Deutscher Verlag der Wissenschaften, Berlin, 1968. · Zbl 0143.27002 [3] M. Křížek: An equilibrium finite element method in three-dimensional elasticity. Apl. Mat. 27 (1982), 46-75. [4] A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Prague, 1977. [5] J. Nečas: Les Méthodes Directes en Théorie des Equations Elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
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