Higher regularity for minimizers of very degenerate convex integrals
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913181159391232 |
|---|---|
| author | Grimaldi, Antonio Giuseppe |
| author_facet | Grimaldi, Antonio Giuseppe |
| contents | In this paper, we consider minimizers of integral functionals of the type
\begin{equation*}
\mathcal{F}(u):= \int_Ω\dfrac{1}{p} \bigl( |Du(x)|_{γ(x)}-1\bigr)_+^p \ \mathrm{d}x, \end{equation*} for $p >1$, where $u : Ω\subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function. Here, $| \cdot |_γ$ is the associated norm of a bounded, symmetric and coercive bilinear form on $\mathbb{R}^{nN}$. We prove that $\mathcal{K}(x,Du)$ is continuous in $Ω$, for any continuous function $\mathcal{K}: Ω\times \mathbb{R}^{nN} \rightarrow \mathbb{R}$ vanishing on $\bigl\{ (x,ξ) \in Ω\times \mathbb{R}^{nN} : |ξ|_{γ(x)} \le 1 \bigr\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17665 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher regularity for minimizers of very degenerate convex integrals Grimaldi, Antonio Giuseppe Analysis of PDEs In this paper, we consider minimizers of integral functionals of the type \begin{equation*} \mathcal{F}(u):= \int_Ω\dfrac{1}{p} \bigl( |Du(x)|_{γ(x)}-1\bigr)_+^p \ \mathrm{d}x, \end{equation*} for $p >1$, where $u : Ω\subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function. Here, $| \cdot |_γ$ is the associated norm of a bounded, symmetric and coercive bilinear form on $\mathbb{R}^{nN}$. We prove that $\mathcal{K}(x,Du)$ is continuous in $Ω$, for any continuous function $\mathcal{K}: Ω\times \mathbb{R}^{nN} \rightarrow \mathbb{R}$ vanishing on $\bigl\{ (x,ξ) \in Ω\times \mathbb{R}^{nN} : |ξ|_{γ(x)} \le 1 \bigr\}$. |
| title | Higher regularity for minimizers of very degenerate convex integrals |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.17665 |