Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26
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Plane harmonic functions
Definition
Let be open, and write for the real coordinates on . A real-valued function is harmonic on when is of class and satisfies Laplace's equation
throughout .
Remarks
This page is plane-specific: the Laplacian is the two-variable operator , and later pages generalize the theory to higher dimensions.
The function is real-valued by convention. Complex-valued harmonic maps are handled componentwise by asking both real coordinates to be harmonic.
Used by
- A harmonic function can vanish on a line without being zero everywhere Counterexample
- The product of two harmonic functions need not be harmonic Counterexample
- Chartwise harmonic and subharmonic functions on a Riemann surface Definition
- Green function with a pole at infinity Definition
- Harmonic conjugates Definition
- Harmonic Hardy classes on the unit disc Definition
- The canonical Green kernel of a plane domain Definition
- An irregular puncture does not force the Green kernel to vanish Example
- Capacity of a disc and its circular equilibrium measure Example
- Finite and countable planar sets have zero logarithmic capacity Example
- Green kernel of the disc at a nonzero pole Example
- Harmonic measure of the two annulus boundary circles Example
- Infinity-pole Green function recovered from a circular conductor Example
- log|z| is harmonic on the punctured plane Example
- A dipole Green function exists on a Riemann surface Lemma
- A planar barrier forces the regularized Perron envelope to have the prescribed boundary limit Lemma
- A simply connected Greenian Riemann surface is a disc Lemma
- A simply connected surface without a Green kernel is plane or sphere Lemma
- Green envelope dichotomy, logarithmic pole and leastness on a Riemann surface Lemma
- Green's second identity on a compact bordered domain of a Riemann surface Lemma
- Harmonic conjugates and integral logarithmic-pole monodromy on surfaces Lemma
- Locality of subharmonicity in the plane and on Riemann surfaces Lemma
- Locally bounded harmonic families have harmonic subsequential limits Lemma
- Logarithmic modulus is harmonic off its centre Lemma
- Maximum principle for a compact logarithmic potential Lemma
- Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
- Removing a compact chart disc gives a Greenian surface Lemma
- Symmetry of the canonical surface Green kernel Lemma
- This page uses the standard upper-semicontinuous subharmonic convention Remark
- A bounded harmonic function near an isolated puncture extends harmonically Theorem
- A plane domain is homologically simply connected exactly when every harmonic function has a global conjugate Theorem
- Borel harmonicity and comparison of harmonic measure Theorem
- Boundary values and zeros of a Blaschke product Theorem
- Canonical Green kernels are unique, symmetric and domain monotone Theorem
- Conformal covariance of the canonical planar Green kernel Theorem
- Every plane harmonic function is locally the real part of a holomorphic function Theorem
- Existence and uniqueness of harmonic measure on a bounded regular plane domain Theorem
- Green function at infinity from the equilibrium potential Theorem
- Green functions exist on all bounded plane domains Theorem
- Green kernel of a simply connected plane domain from a Riemann map Theorem
…and 7 more results.
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
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)