Math 2.0

Potential theory Definition

Least harmonic majorant

Builds on Harmonic majorantHarmonic functionSubharmonic functionOpen setPoisson integral formula

Definition

Let n≥2n \ge 2, let Ω⊆ℝn\Omega \subseteq \R^n be an open set, and let u ⁣:Ω→[−∞,∞)u \colon \Omega \to [-\infty, \infty) be a function. A least harmonic majorant of uu is a harmonic majorant h0h_0 of uu such that h0≤hh_0 \le h on Ω\Omega for every harmonic majorant hh of uu.

Remarks

A least harmonic majorant is unique when it exists, since two of them are each at most the other. If Ω\Omega is connected and uu is subharmonic with at least one harmonic majorant, then uu has a least harmonic majorant. The greatest harmonic minorant of a function vv is defined in the same way; it is −h0-h_0, where h0h_0 is the least harmonic majorant of −v-v.

Example: on the unit disc 𝔻⊆ℂ\mathbb{D} \subseteq \C, the least harmonic majorant of u(z)=∣z∣2u(z) = \abs{z}^2 is the constant 11. It is a harmonic majorant. Let hh be any harmonic majorant, and let ∣a∣<r<1\abs{a} < r < 1. Since h≥r2h \ge r^2 on the circle of radius rr, the Poisson integral formula on that circle gives h(a)≥r2h(a) \ge r^2. Letting r→1r \to 1 gives h(a)≥1h(a) \ge 1.

Non-example: the constant 22 is a harmonic majorant of ∣z∣2\abs{z}^2 on 𝔻\mathbb{D}, but it is not least, because 11 is also a harmonic majorant and 2≤12 \le 1 fails.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

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Continuous function, concept, no evidence levelContinuous functionconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHarmonic majorant, concept, no evidence levelHarmonic majorantconceptInfimum, concept, no evidence levelInfimumconceptOpen ball, concept, no evidence levelOpen ballconceptOpen set, concept, no evidence levelOpen setconceptPoisson kernel, concept, no evidence levelPoisson kernelconceptSubharmonic function, concept, no evidence levelSubharmonic functionconceptSupremum, concept, no evidence levelSupremumconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptJordan measure, definition, no evidence levelJordan measuredefinitionDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionGradient, concept, no evidence levelGradientconceptPartial derivative, definition, no evidence levelPartial derivativedefinitionDivergence, concept, no evidence levelDivergenceconceptDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionDouble integral, definition, no evidence levelDouble integraldefinitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionLaplacian, concept, no evidence levelLaplacianconceptIterated derivative, definition, no evidence levelIterated derivativedefinitionHarmonic function, concept, no evidence levelHarmonic functionconceptC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionParametrised surface, definition, no evidence levelParametrised surfacedefinitionTangent plane and normal vector of a surface, definition, no evidence levelTangent plane and nor…definitionSurface area, definition, no evidence levelSurface areadefinitionPoisson integral formula, statement, no evidence levelPoisson integral form…statementLeast harmonic majorant, definition, no evidence levelLeast harmonic majora…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Continuous function concept
  2. Euclidean space object
  3. Function concept
  4. Harmonic majorant concept
  5. Infimum concept
  6. Open ball concept
  7. Open set concept
  8. Poisson kernel concept
  9. Subharmonic function concept
  10. Supremum concept
  11. Fréchet derivative definition
  12. Total derivative concept
  13. Function of several variables concept
  14. Vector-valued function concept
  15. Jordan measure definition
  16. Differentiable function of several variables concept
  17. Limit of a function of several variables concept
  18. Partial derivative concept
  19. Vector field concept
  20. Continuity in R^n definition
  21. Differentiable function of several variables definition
  22. Gradient concept
  23. Partial derivative definition
  24. Divergence concept
  25. Differentiable on an open set definition
  26. Double integral definition
  27. Gradient definition
  28. Higher partial derivatives definition
  29. Laplacian concept
  30. Iterated derivative definition
  31. Harmonic function concept
  32. C^k function characterised by continuous derivatives statement
  33. Parametrised curve definition
  34. Parametrised surface definition
  35. Tangent plane and normal vector of a surface definition
  36. Surface area definition
  37. Poisson integral formula statement
  38. Least harmonic majorant definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Subharmonic_function
About this page
TagQE4X5PTWN2
URIm2://QE4X5PTWN2
Kinddefinition
WritingW0 stub
MSC31B05, 31A05
DefinesLeast harmonic majorant
UsesHarmonic majorant, Harmonic function, Subharmonic function, Open set, Poisson integral formula

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