A second-order linear elliptic 偏微分 equation that describes systems in a steady-state or equilibrium condition. It is written as [latex]nabla^2 u = 0[/latex] or [latex]Delta u = 0[/latex], where [latex]nabla^2[/latex] (or [latex]Delta[/latex]) is the Laplace operator. Solutions, called harmonic functions, are the smoothest possible functions and represent potentials in fields like electrostatics, gravitation, and fluid flow.
Laplace’s equation is the canonical elliptic PDE. It arises in numerous physical contexts where a quantity is in equilibrium and its value at a point is the average of its values in the surrounding neighborhood. This averaging property is a defining characteristic of its solutions, known as harmonic functions. A direct consequence is the ‘maximum principle’ for harmonic functions, which states that a non-constant solution cannot attain its maximum or minimum value in the interior of its domain; these extrema must lie on the boundary. This prevents, for example, a hot spot from existing in a region of steady-state heat flow unless there is a source there (which would violate [latex]nabla^2 u = 0[/latex]).
Solutions to Laplace’s equation are infinitely differentiable (analytic) even if the boundary conditions are not. This is a remarkable smoothing property, even stronger than that of the 热方程. The problem of finding a solution to Laplace’s equation in a domain given the values of the solution on the boundary is known as the Dirichlet problem. The related Neumann problem specifies the normal derivative on the boundary.
Unlike the time-dependent heat and wave equations, Laplace’s equation is typically solved for boundary value problems, where the entire boundary of a spatial domain influences the solution at every interior point simultaneously. This ‘global’ dependence contrasts with the causal, time-marching nature of parabolic and hyperbolic equations.
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