Overview of Potential Theory: Potential theory studies harmonic and subharmonic functions, Newtonian potentials, and the behavior of physical fields such as gravity and electrostatics. It connects to complex analysis, probability (via Brownian motion), and partial differential equations. Core ideas include harmonic functions, Laplace's equation, Green's functions, and Poisson integrals, with applications to solving boundary value problems and understanding concepts like capacity and equilibrium measures.

Core equations in potential theory include the Laplace equation ∇²φ = 0, which characterizes harmonic potentials in charge-free regions, and Poisson's equation ∇²φ = −ρ/ε₀, which relates the potential to a charge distribution ρ. The fundamental solution in three dimensions is G(x,y) = 1/(4π|x−y|), leading to the Newtonian potential φ(x) = ∫ G(x,y)ρ(y) dy for appropriate domains. Boundary value problems such as Dirichlet and Neumann problems specify the potential or its normal derivative on the boundary and are often solved via Green's identities and boundary integral representations. Green's formula expresses φ(x) in terms of boundary data: φ(x) = ∮ G(x,y) ∂n'φ(y) − φ(y) ∂n'G(x,y) dS(y). The maximum principle states that a harmonic function attains its extrema on the boundary, while energy minimization principles (e.g., minimizing ∫|∇φ|² dV subject to boundary conditions) characterize physically relevant solutions. Gauss's law ∇·E = ρ/ε₀ links the electric field to charge density, and with E = −∇φ it ties the potential to observable fields. In potential theory, these equations underpin boundary value problems and the representation of solutions via Green's functions.

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