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When we consider a partial differential equation, we need to give some boundary condition such that this question is reasonable. What criterion can we use? e.g., heat equation $\partial_t u(t,x) = \Delta_x u(t,x)$ we take Laplace transfrom, then we get $\tau u(\tau,x) - u(0,x) = \Delta u(\tau,x)$. It seems that a boundary condition in the form of initial condtion $u(0,x) = f(x)$ is more reasonable. But is there a general criterion in judging the rationality (or appropriteness) of the boundary conditions?

The Pointer
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Ailiy Evan
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    Some of this comes down to well-posedness theory, which is actually rather complicated. For example, why are sensible boundary conditions for the (1+1)D wave equation different from those of the 2D Laplace equation? This comes down to classification of second order linear PDE, which is a rich subject whose terminology (parabolic/elliptic/hyperbolic) is seen in higher order and/or nonlinear PDE as well. – Ian Feb 05 '21 at 15:52
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    (Cont.) Within the class of well-posed problems, you can ask about meaning in the underlying applied context. For example, there is thermodynamic theory telling you that the boundary conditions for the heat equation should really be Robin if the system is in a heat bath and close to equilibrium, and so Dirichlet or Neumann conditions are really an idealization. – Ian Feb 05 '21 at 15:52
  • Thanks for your helpful comment. – Ailiy Evan Feb 06 '21 at 11:14

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The paper of Kurt Friedrichs, Symmetric Positive Linear Differential Equations, Communications in Pure and Applied Mathematics, vol 11, p333-458 (1958), discusses this question.

Bob Terrell
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