Nemytskii operator

In mathematics, Nemytskii operators are a class of nonlinear operators on L^p 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 (an open and connected set) in n-dimensional Euclidean space. A function f : Ω × R^m → R is said to satisfy the Carathéodory conditions if

• f(x, u) is a continuous function of u for almost all x ∈ Ω;
• f(x, u) is a measurable function of x for all u ∈ R^m.

Given a function f satisfying the Carathéodory conditions and a function u : Ω → R^m, define a new function F(u) : 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 ∈ L^q(Ω; R), 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 L^p(Ω; R^m) into L^q(Ω; R).

References

• (Section 10.3.4)