Math 2.0

Potential theory Definition

Resolutivity of boundary functions

Builds on Open setBoundarySuperharmonic functionSubharmonic functionHarmonic functionDirichlet problem

Definition

Let n≥2n \ge 2, let Ω⊆ℝn\Omega \subseteq \R^n be a bounded open set with boundary ∂Ω\partial\Omega, and let f ⁣:∂Ω→[−∞,+∞]f \colon \partial\Omega \to [-\infty, +\infty] be a function.

  1. The upper class 𝒰f\mathcal{U}_f is the set of functions uu on Ω\Omega with these properties. On each connected component of Ω\Omega, uu is superharmonic or identically +∞+\infty. The function uu is bounded below, and lim inf⁡x→ζ, x∈Ωu(x)≥f(ζ)\liminf_{x \to \zeta,\, x \in \Omega} u(x) \ge f(\zeta) for every ζ∈∂Ω\zeta \in \partial\Omega.
  2. The lower class ℒf\mathcal{L}_f is the set of functions vv on Ω\Omega with these properties. On each connected component of Ω\Omega, vv is subharmonic or identically −∞-\infty. The function vv is bounded above, and lim sup⁡x→ζ, x∈Ωv(x)≤f(ζ)\limsup_{x \to \zeta,\, x \in \Omega} v(x) \le f(\zeta) for every ζ∈∂Ω\zeta \in \partial\Omega.
  3. The upper solution is H‾f(x)=inf⁡{u(x):u∈𝒰f}\overline{H}_f(x) = \inf\set{u(x) : u \in \mathcal{U}_f} and the lower solution is H‾f(x)=sup⁡{v(x):v∈ℒf}\underline{H}_f(x) = \sup\set{v(x) : v \in \mathcal{L}_f}, for x∈Ωx \in \Omega.

The function ff is resolutive if H‾f=H‾f\overline{H}_f = \underline{H}_f on Ω\Omega and this common function is harmonic on Ω\Omega, in particular finite. The common function is then written HfH_f.

Remarks

The inequality H‾f≤H‾f\underline{H}_f \le \overline{H}_f always holds, by the maximum principle applied to v−uv - u for v∈ℒfv \in \mathcal{L}_f and u∈𝒰fu \in \mathcal{U}_f. Wiener's theorem says that every continuous ff is resolutive, so HfH_f is a generalised solution of the Dirichlet problem. For a bounded domain, Brelot showed that ff is resolutive if and only if ff is integrable with respect to harmonic measure. Then Hf(x)H_f(x) is the integral of ff against the harmonic measure for xx.

Example: let Ω\Omega be the unit disc in ℝ2\R^2 and ff continuous on the unit circle. The Poisson integral P[f]P[f] is harmonic in Ω\Omega and tends to f(ζ)f(\zeta) at each boundary point ζ\zeta. So P[f]P[f] lies in both 𝒰f\mathcal{U}_f and ℒf\mathcal{L}_f, which gives H‾f≤P[f]≤H‾f\overline{H}_f \le P[f] \le \underline{H}_f. With H‾f≤H‾f\underline{H}_f \le \overline{H}_f this forces H‾f=H‾f=P[f]\overline{H}_f = \underline{H}_f = P[f], which is harmonic, so ff is resolutive.

Non-example: let f≡+∞f \equiv +\infty on ∂Ω\partial\Omega. The only member of 𝒰f\mathcal{U}_f is the constant +∞+\infty, so H‾f≡+∞\overline{H}_f \equiv +\infty. Every constant cc lies in ℒf\mathcal{L}_f, so H‾f≡+∞\underline{H}_f \equiv +\infty too. The two solutions agree but are not harmonic, so ff is not resolutive.

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  1. Boundary concept
  2. Continuous function concept
  3. Domain in the complex plane concept
  4. Euclidean space object
  5. Function concept
  6. Open set concept
  7. Subharmonic function concept
  8. Superharmonic function concept
  9. Total derivative concept
  10. Function of several variables concept
  11. Vector-valued function concept
  12. Differentiable function of several variables concept
  13. Partial derivative concept
  14. Vector field concept
  15. Gradient concept
  16. Divergence concept
  17. Laplacian concept
  18. Harmonic function concept
  19. Dirichlet problem problem
  20. Resolutivity of boundary functions definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Perron_method
About this page
TagETH7FW1KKA
URIm2://ETH7FW1KKA
Kinddefinition
WritingW0 stub
MSC31B20, 31A25, 31B05
DefinesResolutivity of boundary functions
UsesOpen set, Boundary, Superharmonic function, Subharmonic function, Harmonic function, Dirichlet problem

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