Potential Theory

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4 domain-specific abstractions whose origin domain is Potential Theory.

• Capacity of a set — Assign a potential-theoretic size to a set through an extremal energy or admissible-function problem, detecting thinness and polar sets beyond additive volume.
• Harmonic measure — A boundary probability measure giving the likelihood that Brownian motion started inside a domain first exits through each boundary subset, equivalently representing solutions of the Dirichlet problem.
• Neumann–Poincaré operator — A boundary integral operator built from the normal derivative of the Laplace fundamental solution and used to reduce harmonic boundary-value problems to Fredholm integral equations.
• Subharmonic function — An upper-semicontinuous function whose value at each point is no greater than the average over every sufficiently small surrounding sphere or ball.