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Plane harmonic functions satisfy the mean-value property
Statement
Every plane harmonic function satisfies both the circle and disc mean-value properties of The circle and disc mean-value properties.
Facts & Assumptions
Given: A harmonic function on an open set , a point , and a radius with .
Every open disc is star-shaped and therefore homologically simply connected, and every harmonic function on such a domain is the real part of a holomorphic function (Star-shaped plane domains are homologically simply connected, Harmonic conjugates exist on homologically simply connected plane domains).
A holomorphic function equals its average on every smaller concentric circle (A holomorphic function equals its average on every circle inside a larger concentric holomorphy disc).
Proof
Because the closed disc lies in the open set , choose with . The restriction of to this disc is harmonic, so [L1] gives a holomorphic function on with there.
For every , applying [L2] to on the circle of radius and taking real parts gives
The circle formula of the definition is step 2.1 at .
Multiplying the identity of step 2.1 by and integrating from to gives the disc formula of the definition, because .
Depends on
Used by
- Chartwise harmonic and subharmonic functions on a Riemann surface Definition
- The Nevanlinna class on the disc Definition
- Capacity of a disc and its circular equilibrium measure Example
- Poisson extension of an indicator arc Example
- A harmonic majorant of log^+|F| exists exactly when the radial log^+ means are bounded Lemma
- Blaschke factorization of a Nevanlinna-class function Lemma
- Plane subharmonicity is invariant under biholomorphic change of coordinate Lemma
- Radial p-means of a holomorphic function are nondecreasing Lemma
- Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
- A continuous plane function with the local mean-value property is harmonic Theorem
- An increasing harmonic sequence converges locally uniformly to a harmonic limit or diverges to +infinity Theorem
- Positive harmonic boundary measures and compact normalized families Theorem
- Positive harmonic functions on a disc satisfy Harnack's inequality Theorem
- Subharmonicity is equivalent to harmonic comparison on compactly contained discs Theorem
- The regularized Perron envelope is harmonic Theorem
Dependency tree · two levels
28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jeremy Orloff, MIT 18.04 Topic 5: Introduction to Harmonic Functions (standard reference, not scraped)
- Sigurdur Helgason, MIT 18.112 Lecture 16: Harmonic Functions (standard reference, not scraped)