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.

be non-empty sets, then

— sets of mappings from

The Nemytskii superposition operator

is the mapping induced by the function

, and such that for any function

φ ∈

its image is given by the rule

φ ) ( x ) = h ( x , φ ( x ) ) ∈

The function

is called the generator of the Nemytskii operator

Let Ω be a domain (an open and connected set) in n-dimensional Euclidean space.

A function f : Ω × Rm → R is said to satisfy the Carathéodory conditions if Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, define a new function F(u) : Ω → R by The function F is called a Nemytskii operator.

Suppose that

for any function

Under these conditions the operator

is Lipschitz continuous if and only if there exist functions

Let Ω be a domain, let 1 < p < +∞ and let g ∈ Lq(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 Lp(Ω; Rm) into Lq(Ω; R).