A specification that a function or field takes prescribed values on the boundary of a spatial domain; in boundary value problems this fixes the trace of the solution on the domain's boundary.

Principle

By prescribing boundary values u|∂Ω=g one removes boundary degrees of freedom and in conjunction with an appropriate linear elliptic operator yields well-posedness (existence and uniqueness) when compatibility conditions hold.

Demonstration

For Laplace's equation Δu=0 on a bounded domain Ω, a Dirichlet condition prescribes u(x)=g(x) for x∈∂Ω; solving the boundary value problem finds a harmonic function in Ω matching g on ∂Ω.

Misapplication

Imposing Dirichlet values in the interior rather than strictly on the boundary, or using boundary data inconsistent with the operator (e.g., incompatible traces), produces ill-posed or unsolvable problems.

Consequence

Correctly applied Dirichlet conditions convert differential problems into boundary value problems with unique solutions under standard ellipticity and regularity assumptions; they also determine Green's functions adapted to those boundary values.

Reversal

Neumann boundary conditions fix normal derivatives on the boundary rather than the function values; mixed (Robin) conditions impose linear combinations of value and normal derivative.

Boundary

Applies where a notion of boundary trace is defined (domains with sufficient regularity); it excludes specifications of derivative data (Neumann), integral constraints over the domain, or conditions on interior submanifolds unless reinterpreted as boundary data.

Semantic Tension

Dirichlet versus Cauchy data: Dirichlet prescribes values on the boundary only, while Cauchy data prescribes both values and normal derivatives and may overdetermine elliptic problems.

Synthesis

A Dirichlet boundary condition fixes the value of the unknown on the domain boundary, removing boundary freedom and, for suitable operators and compatible data, producing well-posed boundary value problems and boundary‑adapted solution kernels.