L1kontraktion

L1kontraktion refers to the contraction property in the L1 norm, particularly in the context of operators or evolution equations acting on L1 spaces. In functional analysis, an operator T mapping L1 to itself is called an L1-contraction if for all x and y in L1,

||Tx − Ty||1 ≤ ||x − y||1.

Sometimes a strict contraction is stated with a constant c < 1 so that ||Tx − Ty||1 ≤ c||x − y||1 for all x, y.

In the semigroup setting, a family {S(t)}t≥0 of operators on L1 is said to be L1-contractive if for every t ≥ 0 and all initial data u0, v0 in L1,

||S(t)u0 − S(t)v0||1 ≤ ||u0 − v0||1.

This property implies stability with respect to initial data and often leads to uniqueness of solutions for associated evolution problems.

Applications and significance include the study of evolution equations and partial differential equations. L1-contraction is a central feature of entropy solutions to scalar conservation laws, where the L1 distance between two solutions cannot increase in time, yielding uniqueness and continuous dependence on initial data. It also appears in linear and nonlinear diffusion and transport problems, where semigroups generated by dissipative operators are typically L1-contractive.

Examples include the heat equation on Rn, whose heat semigroup e^{tΔ} is L1-contractive, and many Markov semigroups that preserve mass and positivity. The concept is linked to the contraction mapping principle and to nonlinear semigroup theory, notably the Crandall–Liggett framework for dissipative operators.

See also contraction mapping principle, Crandall–Liggett theorem, entropy solution, semigroup.