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 Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant.
One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. These convergence theorems can often be used to prove existence of harmonic functions having particular properties.
Read more about this topic: Potential Theory
Famous quotes containing the word inequalities:
“The only inequalities that matter begin in the mind. It is not income levels but differences in mental equipment that keep people apart, breed feelings of inferiority.”
—Jacquetta Hawkes (b. 1910)
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—Henry David Thoreau (1817–1862)