Nemytskii operator

(Redirected from Superposition 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

edit

Let   be non-empty sets, then   — sets of mappings from   with values in   and   respectively. The Nemytskii superposition operator   is the mapping induced by the function  , and such that for any function   its image is given by the rule   The function   is called the generator of the Nemytskii operator  .

Definition of Nemytskii operator

edit

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.

Theorem on Lipschitzian Operators

edit

Suppose that  ,   and

 

where 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

edit

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).

References

edit
  • Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 370. ISBN 0-387-00444-0. (Section 10.3.4)
  • Matkowski, J. (1982). "Functional equations and Nemytskii operators". Funkcial. Ekvac. 25 (2): 127–132.