Subharmonic function: Difference between revisions

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Subharmonic functions can be defined on an arbitrary [[Riemannian manifold]].

''Definition:'' Let ''M'' be a Riemannian manifold, and <math>f:\; M \to {\Bbbmathbb R}</math> an [[upper semicontinuous]] function. Assume that for any open subset <math>U\subset M</math>, and any [[harmonic function]] ''f₁'' on ''U'', such that <math>f_1\geq f</math> on the boundary of ''U'', the inequality <math>f_1\geq f</math> holds on all ''U''. Then ''f'' is called ''subharmonic''.

This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality <math>\Delta f\geq 0</math>, where <math>\Delta</math> is the usual [[Laplace–Beltrami_operator|Laplacian]].<ref>{{Cite journal