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

Biagio Ricceri

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arXiv · 2512.08347

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

Biagio Ricceri

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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.

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Biagio Ricceri. 2025-12-09. A constrained approximation theorem for integral functionals on $L^p$. https://arxiv.org/abs/2512.08347

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