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On Nemyckii operators and integral representation of local functionals. (English) Zbl 0536.47027
In this paper, given a measure space $$(\Omega,\mathfrak F,\mu)$$ and a functional $$F(u,B)$$ defined for $$\mu$$-measurable functions $$u$$ and $$\mu$$-measurable sets $$B$$, we study the problem of representing $$F$$ in the integral form $F(u,B)=\int_{B}f(x,u(x))\,d\mu(x)$ for a suitable integrand $$f$$. The main result of the paper is the following.
Suppose that $$F$$ satisfies the following conditions $$(1\leq p<+\infty)$$:
1) $$F(\cdot,\Omega)$$ is lower semicontinuous with respect to the strong $$L^ p(\Omega,\mathbb R^ n)$$ topology;
2) $$F$$ is local on $${\mathfrak F}$$ (i.e. $$u|_ B=v|_ B \mu$$-a.e. $$\Rightarrow F(u,B)=F(v,B));$$
3) $$F(u,\cdot)$$ is finitely additive on $$\mathfrak F$$ for every $$u\in L^ p(\Omega;{\mathbb R}^ n);$$
4) $$F(0,B)=0$$ for every $$B\in {\mathfrak F}$$.
Then we have $$F(u,B)=\int_{B}f(x,u(x))\,d\mu(x),$$ where $$f(x,s)$$ is a measurable function, lower semicontinuous in $$s$$, and such that $$f(x,s)\geq -[a(x)+b| s|^ p]$$ $$(a\in L^ 1(\Omega)$$, $$b\geq 0).$$
In Section 3 we show by a counterexample that hypothesis 1) cannot be dropped.
An interesting consequence of our result is the following.
Let $$T:L^ p(\Omega,{\mathbb R}^ n)\to L^ q(\Omega,{\mathbb R}^ m)$$ be a continuous mapping which is locally defined (i.e. $$u=v$$ $$\mu$$-a.e. on $$B\Rightarrow Tu=Tv$$ $$\mu$$-a.e. on $$B$$). Then $$T$$ is a Nemyckii operator, that is $$(Tu)(x)=f(x,u(x)),$$ where $$f$$ is a Carathéodory function such that $$| f(x,s)| \leq a(x)+b| s|^{p/q}$$ $$(a\in L^ q(\Omega), b\geq 0)$$.

##### MSC:
 47B38 Linear operators on function spaces (general) 46G10 Vector-valued measures and integration 47Gxx Integral, integro-differential, and pseudodifferential operators