A constrained approximation theorem for integral functionals on $L^p$

TL;DR

This paper establishes a constrained approximation theorem for integral functionals on $L^p$ spaces, showing how to approximate functions within a hyperplane while matching integral values under general conditions.

Contribution

It introduces a new approximation result for integral functionals on $L^p$ spaces constrained to hyperplanes, extending previous theories under broad assumptions.

Findings

Provides a method to approximate functions in hyperplanes with prescribed integral values.

Ensures convergence of approximations to a target function while matching integral functionals.

Extends approximation theory for integral functionals in Banach space settings.

Abstract

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\geq 1$. Given a sequence of functions $f, f_1, f_2,...$ from $T\times E$ to ${\bf R}$, under general assumptions, we prove that, for each closed hyperplane $V$ of $L^p(T,E)$, for each $u\in V$, and for each sequence $\{\lambda_n\}$ converging to $\int_Tf(t,u(t))d\mu$, there exists a sequence $\{u_n\}$ in $V$ converging to $u$ and such that $\int_Tf_n(t,u_n(t))d\mu=\lambda_n$ for all $n$ large enough.