Dirichletprobleme

Dirichletprobleme (Dirichlet problem) refers to a class of boundary value problems for partial differential equations, most commonly for Laplace's equation Δu = 0. Given a bounded domain Ω in R^n with boundary ∂Ω and a prescribed boundary function g on ∂Ω, the Dirichlet problem asks for a function u defined on Ω that extends continuously to the boundary and satisfies u = g on ∂Ω and Δu = 0 in Ω. More generally, one may seek solutions to Lu = f in Ω with u = g on ∂Ω for a linear elliptic operator L, often in divergence form Lu = ∇·(A(x)∇u) with boundary condition u = g on ∂Ω.

In the harmonic case, the maximum principle implies that a solution, if it exists, is unique. Existence of a solution for continuous boundary data on a bounded domain is guaranteed by potential-theoretic methods (Perron method, Green's function) and by variational approaches using Sobolev spaces. For nonzero boundary data, the common approach is to reduce to the homogeneous case by extending the boundary data harmonically or by solving a corresponding Dirichlet problem for the interior.

In special geometries, such as the unit disk, explicit formulas express u via the Poisson kernel: u(r,θ) = ∫ g(φ) P_r(θ−φ) dφ. In general, solutions can be represented using Green's function or harmonic measure. Regularity results say that if ∂Ω is smooth and g is smooth, u is smooth up to the boundary; with rough boundaries, regularity may fail. Solutions typically lie in Sobolev spaces H^1(Ω) and gain higher regularity under additional assumptions.

History: the problem is named after Peter Gustav Lejeune Dirichlet and is foundational in potential theory and the study of harmonic functions.