The Dirichlet problem for Laplace's equation consists of finding a solution on some domain such that on the boundary of is equal to some given function. Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
the dirichlet problem for laplace's equation consists of finding a solution on some domain such that on the boundary of is equal to some given function since the laplace operator appears in the heat equation one physical interpretation of this problem is as follows fix the temperature on the boundary of the domain according to the given specification of the boundary condition allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore the temperature distribution in the interior will then be given by the solution to the corresponding dirichlet problem