Maximum Principle

A class of results for certain scalar elliptic and parabolic partial differential equations asserting that a solution cannot attain a nontrivial maximum (or minimum) in the interior of a domain unless it is constant; extrema are constrained to the boundary under the operator's sign conditions.

A family of comparison theorems for elliptic and parabolic partial differential operators asserting that, under appropriate sign and coefficient conditions, a solution attains its maximum (or minimum) on the boundary of the domain or at initial time; strong and weak variants give pointwise and distributional controls.