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.

General definition of Superposition operator

Let ${\textstyle \mathbb {X} ,\ \mathbb {Y} ,\ \mathbb {Z} \neq \varnothing }$ be non-empty sets. Let ${\textstyle \mathbb {Y} ^{\mathbb {X} },\ \mathbb {Z} ^{\mathbb {X} }}$ denote the sets of mappings from ${\textstyle \mathbb {X} }$ to ${\textstyle \mathbb {Y} }$ and ${\textstyle \mathbb {Z} }$ respectively. Let ${\textstyle h\ \colon \mathbb {X} \times \mathbb {Y} \to \mathbb {Z} }$.

Then the Nemytskii superposition operator ${\textstyle H\ \colon \mathbb {Y} ^{\mathbb {X} }\to \mathbb {Z} ^{\mathbb {X} }}$ induced by $h$ is the map taking any map ${\textstyle \varphi \in \mathbb {Y} ^{\mathbb {X} }}$ to the map ${\textstyle H\varphi \in \mathbb {Z} ^{\mathbb {X} }}$ defined by $$(H\varphi )(x)=h(x,\varphi (x))\in \mathbb {Z} ,\quad {\mbox{for all}}\ x\in \mathbb {X} .$$ The function ${\textstyle h}$ is called the generator of the Nemytskii operator ${\textstyle H}$.

Definition of Nemytskii operator

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

$F(u)(x)=f{\big (}x,u(x){\big )}.$

The function F is called a Nemytskii operator.

Theorem on Lipschitzian Operators

Suppose that ${\textstyle h:[a,b]\times \mathbb {R} \to \mathbb {R} }$, ${\textstyle X={\text{Lip}}[a,b]}$ and

$$H:{\text{Lip}}[a,b]\to {\text{Lip}}[a,b]$$

where the operator ${\textstyle H}$ is defined as ${\textstyle \left(Hf\right)\left(x\right)}$ ${\textstyle =h(x,f(x))}$ for any function ${\textstyle f:[a,b]\to \mathbb {R} }$ and any ${\textstyle x\in [a,b]}$. Under these conditions the operator ${\textstyle H}$ is Lipschitz continuous if and only if there exist functions ${\textstyle G,H\in {\text{Lip}}[a,b]}$ such that

$$h(x,y)=G(x)y+H(x),\quad x\in [a,b],\quad y\in \mathbb {R} .$$

Boundedness theorem

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

${\frac {1}{p}}+{\frac {1}{q}}=1.$

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 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. 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.