Minimierer

Minimierer is the term used in German to denote an object that minimizes a given quantity. In mathematics, a minimizer is a function, sequence, or other object that achieves the smallest value of a functional or objective over a specified set. The problem is usually written as: find u in a feasible space X such that J[u] = inf { J[v] : v in X }. The feasible space often includes boundary conditions or additional constraints.

In the calculus of variations, minimizers arise as solutions to problems where a functional assigns a real number to functions. Existence and regularity of minimizers are central questions. The direct method in the calculus of variations provides criteria for existence, typically requiring coercivity (the functional grows at infinity) and lower semicontinuity. Regularity results show that minimizers are often smoother than initially assumed. Classical examples include the Dirichlet energy E(u) = ∫ |∇u|^2, whose minimizers are harmonic functions, and the length functional for curves, whose minimizers are geodesics.

In physics and geometry, minimizers reflect stable or extremal states. The principle of least action states that the physical trajectory minimizes (or makes stationary) the action functional. In geometry, minimizing energy or length leads to important structures such as minimal surfaces and geodesics, which satisfy corresponding Euler–Lagrange equations.

In optimization and numerical analysis, minimizers can be unique if the functional is strictly convex; otherwise multiple local minimizers may exist. Computational methods such as gradient descent, Newton’s method, or proximal algorithms are employed to approximate minimizers, often under constraints via Lagrange multipliers or KKT conditions. Minimizers thus play a foundational role across mathematics, physics, and applied sciences.