Some regularity results for $p$-harmonic mappings between Riemannian manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866929538786656256 |
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| author | Guo, Chang-Yu Xiang, Chang-Lin |
| author_facet | Guo, Chang-Yu Xiang, Chang-Lin |
| contents | Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally $C^{1,α}$ for some $α\in (0,1)$, provided that $N$ is simply connected and has non-positive sectional curvature. We also prove similar results for each minimizing $p$-harmonic mapping $u\colon M\to N$ with $u(M)$ being contained in a regular geodesic ball. Moreover, when $M$ has non-negative Ricci curvature and $N$ is simply connected and has non-positive sectional curvature, we deduce a quantitative gradient estimate for each $C^1$-smooth weakly $p$-harmonic mapping $u\colon M\to N$. Consequently, we obtain a Liouville-type theorem for $C^1$-smooth weakly $p$-harmonic mappings in the same setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_01010 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Some regularity results for $p$-harmonic mappings between Riemannian manifolds Guo, Chang-Yu Xiang, Chang-Lin Differential Geometry Analysis of PDEs 58E15, 49F22 Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally $C^{1,α}$ for some $α\in (0,1)$, provided that $N$ is simply connected and has non-positive sectional curvature. We also prove similar results for each minimizing $p$-harmonic mapping $u\colon M\to N$ with $u(M)$ being contained in a regular geodesic ball. Moreover, when $M$ has non-negative Ricci curvature and $N$ is simply connected and has non-positive sectional curvature, we deduce a quantitative gradient estimate for each $C^1$-smooth weakly $p$-harmonic mapping $u\colon M\to N$. Consequently, we obtain a Liouville-type theorem for $C^1$-smooth weakly $p$-harmonic mappings in the same setting. |
| title | Some regularity results for $p$-harmonic mappings between Riemannian manifolds |
| topic | Differential Geometry Analysis of PDEs 58E15, 49F22 |
| url | https://arxiv.org/abs/1802.01010 |