A constrained approximation theorem for integral functionals on $L^p$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909950976983040 |
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| author | Ricceri, Biagio |
| author_facet | Ricceri, Biagio |
| contents | Let $(T,{\cal F},μ)$ be a $σ$-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 $\{λ_n\}$ converging to $\int_Tf(t,u(t))dμ$, there exists a sequence $\{u_n\}$ in $V$ converging to $u$ and such that $\int_Tf_n(t,u_n(t))dμ=λ_n$ for all $n$ large enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A constrained approximation theorem for integral functionals on $L^p$ Ricceri, Biagio Functional Analysis Classical Analysis and ODEs Let $(T,{\cal F},μ)$ be a $σ$-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 $\{λ_n\}$ converging to $\int_Tf(t,u(t))dμ$, there exists a sequence $\{u_n\}$ in $V$ converging to $u$ and such that $\int_Tf_n(t,u_n(t))dμ=λ_n$ for all $n$ large enough. |
| title | A constrained approximation theorem for integral functionals on $L^p$ |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2512.08347 |