Nemytskii operator


In mathematics, Nemytskii operators are a class of nonlinear operators on Lp spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii.

General definition of Superposition operator

Let be non-empty sets. Let denote the sets of mappings from to and respectively.
Let.
Then the Nemytskii superposition operator induced by is the map taking any map to the map defined by
The function is called the generator of the Nemytskii operator.

Definition of Nemytskii operator

Let Ω be a domain in n-dimensional Euclidean space. A function f : Ω × Rm → R is said to satisfy the Carathéodory conditions iff is a continuous function of u for almost all x ∈ Ω;f is a measurable function of x for all u ∈ Rm.
Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, define a new function F : Ω → R by
The function F is called a Nemytskii operator.

Theorem on Lipschitzian Operators

Suppose that, and
where the operator is defined as for any function and any. Under these conditions the operator is Lipschitz continuous if and only if there exist functions such that

Boundedness theorem

Let Ω be a domain, let 1 < p < +∞ and let g ∈ Lq, with
Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,
Then the Nemytskii operator F as defined above is a bounded and continuous map from Lp into Lq.