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Laplace's equation

The PDE $\nabla^2u=0$ , satisfied by a quantity in steady state with no sources.

Laplace's equation, $\nabla^2u=0$ (one spatial dimension: $u_{xx}=0$ ; two: $u_{xx}+u_{yy}=0$ ), is satisfied by a quantity that has settled into a time-independent (steady-state) distribution with no internal sources.