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.

Definition

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

[math]\displaystyle{ F(u)(x) = f \big( x, u(x) \big). }[/math]

The function F is called a Nemytskii operator.

Boundedness theorem

Let Ω be a domain, let 1 < p < +∞ and let g ∈ L^q(Ω; R), with

[math]\displaystyle{ \frac1{p} + \frac1{q} = 1. }[/math]

Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,

[math]\displaystyle{ \big| f(x, u) \big| \leq C | u |^{p - 1} + g(x). }[/math]

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

• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 370. ISBN 0-387-00444-0.  (Section 10.3.4)