Potential theory

Potential theory may be defined as the study of harmonic functions.

Inequalities

A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the maximum principle. Another important result is Liouville's theorem, which states the only bounded harmonic functions defined on the whole of Rn are, in fact, constant functions. In addition to these basic inequalities, one has such inequalities as Cauchy's estimate, Harnack's inequality, and the Schwarz lemma.

Related Topics:
Maximum principle - Liouville's theorem - Cauchy's estimate - Harnack's inequality - Schwarz lemma

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One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions. These convergence theorems can often be used to prove existence of harmonic functions having particular properties.

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