Poissonproblem

The Poisson problem is a prototypical boundary value problem for Poisson's equation ∆u = f in a domain Ω ⊂ R^n, with conditions specified on the boundary ∂Ω. Here ∆ is the Laplacian, ∆u = sum_i ∂^2u/∂x_i^2, and f represents a source term. It describes steady-state processes where a potential or field u is driven by internal sources.

Common boundary conditions include Dirichlet conditions u = g on ∂Ω, Neumann conditions ∂u/∂n = h on ∂Ω, and Robin conditions αu + β∂u/∂n = r on ∂Ω. For smooth domains and data, classical (smooth) solutions may exist; in general, one seeks weak solutions in appropriate Sobolev spaces.

The problem is often formulated variationally. For Dirichlet data, the weak formulation seeks u ∈ H^1(Ω) with trace on ∂Ω equal to g such that ∫Ω ∇u · ∇v = ∫Ω f v for all v in the homogeneous space H^1_0(Ω). In homogeneous cases, the problem reduces to finding u ∈ H^1_0(Ω) satisfying the same identity. Existence and uniqueness follow from the Lax–Milgram theorem, under suitable assumptions on f and the domain. For Neumann problems, solvability requires a compatibility condition ∫Ω f = ∫∂Ω h, and solutions are determined up to an additive constant.

Numerical methods such as finite elements, finite differences, and finite volumes are standard for approximating solutions. The Poisson problem is central in potential theory and appears across physics and engineering, including electrostatics, heat conduction, fluid mechanics, elasticity, and groundwater flow. Its study encompasses regularity theory, Green’s functions, and various analytical and computational techniques, with the problem named after Siméon Denis Poisson.