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Nemytskii operator

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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

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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

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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

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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

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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

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  • 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.