maximumprinciple

Maximum principle is a collection of results in analysis that bound the behavior of functions solving certain equations, most often partial differential equations or analytic functions. It typically asserts that a function cannot attain an interior extremum unless it is constant, and that extremal values are controlled by boundary data or conditions.

In the setting of harmonic functions, the simplest form states that if u is twice continuously differentiable in a bounded domain D and satisfies Laplace’s equation Δu = 0 in D, and is continuous on the closure of D, then the maximum and minimum of u on the closure occur on the boundary. Consequently, if u achieves its maximum inside D, then u is constant. This is the strong maximum principle, with the weak version also applied to subharmonic and superharmonic functions (Δu ≥ 0 or Δu ≤ 0).

More generally, the maximum principle extends to second-order elliptic operators Lu = ∑ a_ij ∂^2u/∂x_i∂x_j + ∑ b_i ∂u/∂x_i + c u under suitable conditions on the coefficients, showing that a subsolution cannot exceed its boundary values. The principle also yields uniqueness and comparison results for boundary-value problems.

A parabolic version applies to the heat equation u_t − Δu = 0 in a space-time region: the maximum of u in the closure occurs on the parabolic boundary, leading to uniqueness of solutions given initial and boundary data.

In complex analysis, the maximum modulus principle asserts that if f is analytic on a domain, then |f| attains its maximum on the boundary of a compact subset, implying no nonconstant analytic function can have an interior maximum.

The maximum principle provides fundamental a priori estimates and underpins many existence, uniqueness, and stability results across analysis.