Subharmonic function: Difference between revisions

Revision as of 04:18, 3 February 2023

Revision as of 02:18, 23 August 2023

Line 18:

* A function is [[harmonic function|harmonic]] [[if and only if]] it is both subharmonic and superharmonic.

* If <math>\phi</math> is ''C''² ([[smooth function|twice continuously differentiable]]) on an [[open set]] <math>G</math> in <math>\R^n</math>, then <math>\phi</math> is subharmonic [[if and only if]] one has <math> \Delta \phi \geq 0</math> on <math>G</math>, where <math>\Delta</math> is the [[Laplacian]].

* The [[maxima and minima|maximum]] of a subharmonic function cannot be achieved in the [[interior (topology)|interior]] of its ___domain unless the function is constant, thiswhich is the so-called the [[maximum principle]]. However, the [[minimum]] of a subharmonic function can be achieved in the interior of its ___domain.

* Subharmonic functions make a [[convex cone]], that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.

*The [[pointwise maximum]] of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.