Dirichlet boundary condition
/ DEER-ish-lay /
A PDE has infinitely many solutions until you say what is happening at the edges of the region. The simplest way to do that is to dictate the value of the unknown function itself all along the boundary — to say not how steeply the temperature changes at the wall, but exactly what the temperature is there. That prescription is the Dirichlet boundary condition, the first and most intuitive of the standard boundary conditions.
Formally, on the boundary of the domain you require u = g, where g is a given function defined on that boundary (it may be a constant, or vary from point to point along the edge). Physically this is what you impose when the boundary is held at a known state: a rod whose ends are clamped to ice and a furnace (temperature fixed at 0 and at some hot value), a drumhead nailed down at its rim (displacement fixed at zero), an electrode held at a known voltage. A condition where the prescribed value is zero, u = 0 on the boundary, is called homogeneous Dirichlet, and it is the case that makes separation of variables produce clean sine series.
Dirichlet conditions are ubiquitous because so many physical boundaries genuinely hold a fixed value: clamped temperatures, pinned displacements, grounded or energized conductors. For Laplace's and Poisson's equations the Dirichlet problem — prescribe u on the whole boundary, solve inside — is the best-behaved PDE problem there is: it has exactly one solution, that solution depends continuously on the boundary data, and it obeys the maximum principle. They contrast with Neumann conditions, which fix the flux (the normal derivative) instead of the value.
A rod of length L with u(0, t) = 0 and u(L, t) = 0 for all t has homogeneous Dirichlet conditions (both ends held at zero temperature). These force the spatial eigenfunctions to be sin(n pi x / L), which vanish at both ends, giving a pure sine series.
Fixing the value at both ends selects sines; this is the most common setup in introductory PDE problems.
Dirichlet fixes the value, not the slope; confusing it with the Neumann condition (which fixes the normal derivative, i.e. the flux) changes which eigenfunctions arise — sines versus cosines — and can give a completely different answer.