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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2602.05436 |
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| _version_ | 1866917250821259264 |
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| author | Gjokaj, Anton Kalaj, David |
| author_facet | Gjokaj, Anton Kalaj, David |
| contents | We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\toΩ$ between Lyapunov domains (equivalently, bounded $C^{1,α}$ domains) in $\mathbb{R}^n$, $α\in(0,1]$, is globally Lipschitz on $\overline D$. The argument is based on a boundary iteration scheme: an initial Hölder modulus for the boundary trace (coming from quasiconformality) is improved via the $C^{1,α}$ graph representation of $\partialΩ$, yielding higher Hölder regularity for the normal component. This boundary gain is converted into a near-boundary gradient bound for harmonic functions through a basepoint boundary Hölder-to-gradient estimate obtained by flattening the boundary and using local harmonic-measure bounds. Quasiconformality then propagates the resulting control from one component to the full differential, and iteration gives boundedness of $|Df|$ up to the boundary. Along the way we briefly survey several standard tools from the theory of quasiconformal harmonic mappings (QCH), including boundary Hölder continuity, distortion of derivatives, and boundary-to-interior propagation principles that enter the iteration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05436 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lipschitz regularity of harmonic quasiconformal maps between Lyapunov domains in $\mathbb{R}^n$ Gjokaj, Anton Kalaj, David Analysis of PDEs We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\toΩ$ between Lyapunov domains (equivalently, bounded $C^{1,α}$ domains) in $\mathbb{R}^n$, $α\in(0,1]$, is globally Lipschitz on $\overline D$. The argument is based on a boundary iteration scheme: an initial Hölder modulus for the boundary trace (coming from quasiconformality) is improved via the $C^{1,α}$ graph representation of $\partialΩ$, yielding higher Hölder regularity for the normal component. This boundary gain is converted into a near-boundary gradient bound for harmonic functions through a basepoint boundary Hölder-to-gradient estimate obtained by flattening the boundary and using local harmonic-measure bounds. Quasiconformality then propagates the resulting control from one component to the full differential, and iteration gives boundedness of $|Df|$ up to the boundary. Along the way we briefly survey several standard tools from the theory of quasiconformal harmonic mappings (QCH), including boundary Hölder continuity, distortion of derivatives, and boundary-to-interior propagation principles that enter the iteration. |
| title | Lipschitz regularity of harmonic quasiconformal maps between Lyapunov domains in $\mathbb{R}^n$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.05436 |