Dirichlet boundary condition

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

For the forward diffusion equation $u_t=\kappa u_{yy}$ on $0<y<L$, with $\kappa >0$ and homogeneous Dirichlet boundary conditions, a solution continuous on the closed initial-boundary domain can take initial data $u(0,y)=\varphi (y)$ only if $\varphi (0)=\varphi (L)=0$. These equalities express the same corner value through the initial condition and the boundary condition.

An incompatible endpoint value does not prevent an interior or L2 space initial-data solution. For example, on $0<y<1$ the initial ramp $\varphi (y)=y$ givesIt satisfies both zero endpoint values for every $t>0$ and tends to $y$ at every fixed $0<y<1$ as $t\downarrow 0$, also in the L2 norm. It cannot extend continuously to $(0,1)$ with both the initial value one and the boundary value zero. Each sine term vanishes at the endpoint, so the series cannot silently repair that pointwise contradiction.

If the equation is written $a{u}_t=u_{yy}$, the intended forward heat coefficient requires $a>0$. For $a<0$, the ramp's coefficients would grow like $e^{n^2{\pi }^{2}t/|a|}/n$, failing square summability for every $t>0$. This example distinguishes an incompatible corner from the separate ill-posedness of backward diffusion.

Dirichlet boundary data prescribe the boundary values of a solution as a function $\phi$ on the boundary of its domain. Continuity on a polygon requires agreement of the values at corners, but continuity alone does not ensure solvability if the equation loses uniform ellipticity at a boundary side.

In a fixed-temperature convection problem, the boundary temperature is prescribed, and its perturbation satisfies a thermal Dirichlet boundary condition. A homogeneous condition means the boundary follows the prescribed conductive background temperature. This does not necessarily mean a vertical wall has a spatially constant temperature; that additional assumption can conflict with a background vertical gradient.