Math 2.0

Potential theory Dirichlet problem and Green's functions Theorem

Poisson integral formula

Builds on Harmonic functionPoisson kernelOpen ballContinuous functionSurface area

Statement

Let n≥2n \ge 2 and R>0R > 0, let B={x∈ℝn:∣x∣<R}B = \set{x \in \R^n : \abs{x} < R} be the open ball, and let S=∂BS = \partial B be its boundary sphere with surface measure dSdS. Let σn−1\sigma_{n-1} be the surface area of the unit sphere in ℝn\R^n, and for x∈Bx \in B and y∈Sy \in S put P(x,y)=R2−∣x∣2σn−1R ∣x−y∣n.P(x, y) = \frac{R^2 - \abs{x}^2}{\sigma_{n-1} R \, \abs{x - y}^n}.

  1. If uu is continuous on B‾\overline{B} and harmonic on BB, then u(x)=∫SP(x,y) u(y) dS(y)u(x) = \int_S P(x, y) \, u(y) \, dS(y) for every x∈Bx \in B.
  2. If gg is continuous on SS, then u(x)=∫SP(x,y) g(y) dS(y)u(x) = \int_S P(x, y) \, g(y) \, dS(y) is harmonic on BB, and u(x)→g(y0)u(x) \to g(y_0) as x→y0x \to y_0 for every y0∈Sy_0 \in S.

Remarks

Proof idea: P(⋅,y)P(\cdot, y) is harmonic in BB, positive, and has integral 11 over SS. It is minus the outward normal derivative, in the second variable, of the Green's function of the ball. Part 1 follows from Green's representation formula on smaller balls and a limit. Part 2 holds because P(x,⋅)P(x, \cdot) concentrates near y0y_0 as x→y0x \to y_0, so the integral is an approximate identity. For n=2n = 2 and R=1R = 1 the formula is u(reiθ)=12π∫02π1−r21−2rcos⁡(θ−t)+r2 u(eit) dtu(re^{i\theta}) = \frac{1}{2\pi}\int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - t) + r^2} \, u(e^{it}) \, dt, the classical Poisson kernel of the disc. At x=0x = 0 the formula gives the mean value property: u(0)u(0) is the average of uu over SS. Together, parts 1 and 2 solve the Dirichlet problem on a ball, with a unique solution.

Going deeper

For a student

The Poisson formula is the main tool for local results on harmonic functions. From it one gets Harnack's inequality, gradient estimates, the real analyticity of harmonic functions, and the fact that uniform limits of harmonic functions are harmonic. It also yields the converse of the mean value property. It is the basic step in Perron's method, where a function is replaced by its Poisson integral on a small ball.

Reading path

To understand this page from the ground up, read these 34 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 levelFunctionconceptInfimum, concept, no evidence levelInfimumconceptOpen ball, concept, no evidence levelOpen ballconceptPoisson kernel, concept, no evidence levelPoisson kernelconceptSupremum, 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…statement
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. Infimum concept
  5. Open ball concept
  6. Poisson kernel concept
  7. Supremum concept
  8. Fréchet derivative definition
  9. Total derivative concept
  10. Function of several variables concept
  11. Vector-valued function concept
  12. Jordan measure definition
  13. Differentiable function of several variables concept
  14. Limit of a function of several variables concept
  15. Partial derivative concept
  16. Vector field concept
  17. Continuity in R^n definition
  18. Differentiable function of several variables definition
  19. Gradient concept
  20. Partial derivative definition
  21. Divergence concept
  22. Differentiable on an open set definition
  23. Double integral definition
  24. Gradient definition
  25. Higher partial derivatives definition
  26. Laplacian concept
  27. Iterated derivative definition
  28. Harmonic function concept
  29. C^k function characterised by continuous derivatives statement
  30. Parametrised curve definition
  31. Parametrised surface definition
  32. Tangent plane and normal vector of a surface definition
  33. Surface area definition
  34. Poisson integral formula statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_kernel
About this page
TagPCZBE7EX01
URIm2://PCZBE7EX01
Kindstatement, theorem
WritingW0 stub
MSC31B10, 31B20, 31A10
StatusThis copy of the site has no build data. Run m2 status PCZBE7EX01.
AboutPoisson kernel, Dirichlet problem
UsesHarmonic function, Poisson kernel, Open ball, Continuous function, Surface area

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