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.

Definition

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.

Boundedness theorem

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

  • 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)
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