Math 2.0

Potential theory Dirichlet problem and Green's functions Theorem

Poisson integral for the half-space

Builds on Harmonic functionPoisson kernelContinuous functionSurface area

Statement

Let n≥2n \ge 2, write points of ℝn\R^n as x=(x′,xn)x = (x', x_n) with x′∈ℝn−1x' \in \R^{n-1}, and let H={x∈ℝn:xn>0}H = \set{x \in \R^n : x_n > 0} be the upper half-space. Let σn−1\sigma_{n-1} be the surface area of the unit sphere in ℝn\R^n, and for x∈Hx \in H and y∈ℝn−1y \in \R^{n-1} put P(x,y)=2xnσn−1(∣x′−y∣2+xn2)n/2.P(x, y) = \frac{2 x_n}{\sigma_{n-1} \left( \abs{x' - y}^2 + x_n^2 \right)^{n/2}}. Let g ⁣:ℝn−1→ℝg \colon \R^{n-1} \to \R be bounded and continuous, and define u(x)=∫ℝn−1P(x,y) g(y) dy,x∈H.u(x) = \int_{\R^{n-1}} P(x, y) \, g(y) \, dy, \qquad x \in H. Then uu is bounded and harmonic on HH, and u(x)→g(y0)u(x) \to g(y_0) as x→(y0,0)x \to (y_0, 0) with x∈Hx \in H, for every y0∈ℝn−1y_0 \in \R^{n-1}.

Remarks

Proof idea: for fixed yy, P(⋅,y)P(\cdot, y) is harmonic in HH. It is, up to a constant factor, the normal derivative at the boundary of the Green's function of HH built by reflection. The kernel is positive and ∫ℝn−1P(x,y) dy=1\int_{\R^{n-1}} P(x, y) \, dy = 1, so ∣u∣≤sup⁡∣g∣\abs{u} \le \sup \abs{g}. As xn→0x_n \to 0 the kernel concentrates at x′x', which gives the boundary values. For n=2n = 2 the kernel is 1πx2(x1−y)2+x22\frac{1}{\pi} \frac{x_2}{(x_1 - y)^2 + x_2^2}, the Poisson kernel of the upper half-plane. Boundedness matters for uniqueness: u(x)=xnu(x) = x_n is harmonic on HH with boundary values 00, but it is not the Poisson integral of 00.

Going deeper

For a student

The half-space formula is the model for boundary behaviour of harmonic functions, nontangential limits and Hardy spaces. In ℝ2\R^2 it is the link between harmonic functions on the upper half-plane and the Hilbert transform. The family P(⋅,y)P(\cdot, y) for xn=tx_n = t fixed is the Poisson semigroup, an approximate identity used throughout harmonic analysis.

Reading path

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_kernel
About this page
TagWXFTF05MHK
URIm2://WXFTF05MHK
Kindstatement, theorem
WritingW0 stub
MSC31B10, 31B20, 31A10
StatusThis copy of the site has no build data. Run m2 status WXFTF05MHK.
AboutPoisson kernel, Dirichlet problem
UsesHarmonic function, Poisson kernel, Continuous function, Surface area

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