Nemytskii operator

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

F(u)(x) = f \big( x, u(x) \big).

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

\frac1{p} + \frac1{q} = 1.

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

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

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 and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. pp. 356. ISBN 0-387-00444-0.  (Section 9.3.4)

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