Maximum principle

A theorem stating that solutions of certain elliptic equations cannot attain a nonconstant interior maximum or minimum under specified conditions. It links local differential constraints to global bounds determined by a domain’s boundary.

Weak maximum principle: A result bounding a function satisfying an elliptic differential inequality by its values on the boundary. This form gives the basic boundary bound without excluding interior extrema that are not strict.
Uniqueness of solutions: The property that a mathematical problem has at most one solution satisfying its conditions. Boundary comparisons and the maximum principle often show that two candidate solutions must coincide.
Laplacian: The differential operator that sums a function’s second partial derivatives in Euclidean coordinates. The classical maximum principle is most familiar for equations defined by this operator.
Minimum principle: A result restricting where functions satisfying specified equations can attain their minimum. For many elliptic equations, it is the sign-reversed counterpart of the maximum principle.
Strong maximum principle: A result stating that an interior maximum of a suitable elliptic solution forces the solution to be constant. This sharper form rules out nonconstant solutions attaining an interior maximum.
Dirichlet problem: A boundary value problem prescribing a function’s values on the boundary of a domain. The principle bounds solutions by prescribed boundary data and supports uniqueness.
Uniform ellipticity: A condition ensuring an elliptic operator’s principal coefficients remain bounded and nondegenerate. Uniform ellipticity is a common hypothesis guaranteeing strong forms of the principle.
Maximum principle for parabolic equations: A theorem controlling extrema of solutions to parabolic equations over space and time. Its time-dependent setting changes which boundary portions control extrema.
Hopf lemma: A boundary-point result asserting a strict outward normal derivative when a nonconstant elliptic solution reaches an extremum at the boundary. It refines the principle by describing how a solution meets the boundary at an extremum.
Elliptic regularity: The theory describing how elliptic equations force solutions to have smoothness beyond their initial assumptions. Maximum-principle estimates complement regularity results in controlling elliptic solutions.

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