Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.20597 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909709066305536 |
|---|---|
| author | Zhu, Yizhe |
| author_facet | Zhu, Yizhe |
| contents | We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f: \overline{\mathbb Y}\xrightarrow{\rm onto} \overline{\mathbb X}$ where $\mathbb X$ and $\mathbb Y$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0 \colon \partial \mathbb Y \to \partial \mathbb X$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1 < p < \infty$, the functional $\mathcal{E}_p$ admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness of diffeomorphic minimizers of $L^p$-mean distortion Zhu, Yizhe Functional Analysis We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f: \overline{\mathbb Y}\xrightarrow{\rm onto} \overline{\mathbb X}$ where $\mathbb X$ and $\mathbb Y$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0 \colon \partial \mathbb Y \to \partial \mathbb X$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1 < p < \infty$, the functional $\mathcal{E}_p$ admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique. |
| title | Uniqueness of diffeomorphic minimizers of $L^p$-mean distortion |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.20597 |