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
- 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 ∈ Rm.
Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, 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 ∈ Lq(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 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. pp. 370. ISBN 0-387-00444-0. (Section 10.3.4)
Original source: https://en.wikipedia.org/wiki/Nemytskii operator.
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