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Erschienen in:
2017 | OriginalPaper | Buchkapitel
The Potential Equation
verfasst von : Wolfgang Hackbusch
Erschienen in: Elliptic Differential Equations
Verlag: Springer Berlin Heidelberg
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In Section 2.1 the simplest but prototypical elliptic differential equation of second order is presented. The solutions of this equation are called harmonic. Together with a boundary condition, one obtains a boundary-value problem. An important tool is the singularity function, which is defined in Section 2.2. The Green formulae allow a representation of the solution in Theorem 2.8. In Section 2.3 functions with mean-value property are introduced. It is shown that these functions coincide with harmonic functions. The mean-value property implies the maximum minimum principle: non-constant functions have no local extrema. An important conclusion is the uniqueness of the solution (Theorem 2.18). Finally, in Section 2.4, it is shown that the solution depends continuously on the boundary data.