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Maximum principles state that solutions to elliptic and parabolic PDEs achieve their extreme values on the boundary of the domain (or at the initial time for parabolic equations), not in the interior. For harmonic functions (solutions to Laplace's equation), this means the maximum and minimum occur on the boundary. For the heat equation, the maximum of u over a space-time cylinder occurs on the parabolic boundary (the initial time or the spatial boundary). These principles are fundamental tools for proving uniqueness, comparison results, and a priori estimates.
The maximum principle is perhaps the single most important qualitative property of elliptic and parabolic PDEs. In its simplest form for Laplace's equation, it states: if u is harmonic in a connected open set Ω and achieves its maximum at an interior point, then u is constant. The proof uses the mean value property—the value of a harmonic function at any point equals its average over any surrounding sphere—which makes it impossible for a strict interior maximum to exist.
The weak maximum principle says max_Ω u = max_∂Ω u; the strong maximum principle strengthens this to say that if the maximum is achieved in the interior, u must be identically constant. The strong version is considerably more useful: it gives uniqueness of the Dirichlet problem (two solutions with the same boundary data must be identical), continuous dependence on boundary data, and comparison principles (if one solution dominates another on the boundary, it dominates everywhere).
For parabolic equations like the heat equation u_t = Δu, the maximum principle takes a modified form. On a space-time cylinder Q = Ω × (0,T], the maximum of u occurs on the parabolic boundary ∂_p Q = (Ω × {0}) ∪ (∂Ω × [0,T])—the bottom and sides, but not the top. This asymmetry reflects the irreversibility of diffusion: the future is determined by the past, not vice versa. Physically, it says that without internal heat sources, the hottest point is always on the boundary or at the initial time.
Maximum principles extend far beyond the Laplacian. For a general second-order elliptic operator Lu = -aiju_{ij} + bi u_i + cu, the maximum principle holds when c ≥ 0 (no internal sources). The Alexandrov-Bakelman-Pucci (ABP) maximum principle provides quantitative bounds relating the maximum of u to the Ln norm of the right-hand side. These refined maximum principles are essential tools in the regularity theory for nonlinear elliptic and parabolic equations, providing the a priori estimates needed to prove existence of solutions.