Optimal boundary regularity of harmonic maps from $RCD(K,N)$-spaces to $CAT(0)$-spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916758612344832 |
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| author | Zhang, Hui-Chun Zhu, Xi-Ping |
| author_facet | Zhang, Hui-Chun Zhu, Xi-Ping |
| contents | In 1983, Schoen-Uhlenbeck \cite{SU83} established boundary regularity for energy-minimizing maps between smooth manifolds with the Dirichlet boundary condition under the assumption that both the boundary and the data are of $C^{2,α}$. A natural problem is to study the qualitative boundary behavior of harmonic maps with rough boundary and/or non-smooth boundary data.
For the special case where $u$ is a harmonic function on a domain $Ω\subset \mathbb R^n$, this problem has been extensively studied (see, for instance, the monograph \cite{Kenig94}, the proceedings of ICM 2010 \cite{Tor10} and the recent work of Mourgoglou-Tolsa \cite{MT24}). The $W^{1,p}$-regularity ($1<p<\infty$) has been well-established when $\partialΩ$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partialΩ)$. However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples \cite{HL32}, the gradient $|\nabla u|(x)$ may have logarithmic growth as $x$ approaches the boundary $\partial Ω$ even if the boundary is smooth.
In this paper, we first establish a version of the Gauss-Green formula for bounded domains in $RCD(K, N)$ metric measure space. We then apply it to obtain the optimal boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces. Our result is new even for harmonic functions on Lipschitz domains of Euclidean spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal boundary regularity of harmonic maps from $RCD(K,N)$-spaces to $CAT(0)$-spaces Zhang, Hui-Chun Zhu, Xi-Ping Differential Geometry Metric Geometry 58E20, 53C23 In 1983, Schoen-Uhlenbeck \cite{SU83} established boundary regularity for energy-minimizing maps between smooth manifolds with the Dirichlet boundary condition under the assumption that both the boundary and the data are of $C^{2,α}$. A natural problem is to study the qualitative boundary behavior of harmonic maps with rough boundary and/or non-smooth boundary data. For the special case where $u$ is a harmonic function on a domain $Ω\subset \mathbb R^n$, this problem has been extensively studied (see, for instance, the monograph \cite{Kenig94}, the proceedings of ICM 2010 \cite{Tor10} and the recent work of Mourgoglou-Tolsa \cite{MT24}). The $W^{1,p}$-regularity ($1<p<\infty$) has been well-established when $\partialΩ$ is Lipschitz (or even more general) and the boundary data belongs to $W^{1,p}(\partialΩ)$. However, for the endpoint case where the boundary data is Lipschitz continous, as demonstrated by Hardy-Littlewood's classical examples \cite{HL32}, the gradient $|\nabla u|(x)$ may have logarithmic growth as $x$ approaches the boundary $\partial Ω$ even if the boundary is smooth. In this paper, we first establish a version of the Gauss-Green formula for bounded domains in $RCD(K, N)$ metric measure space. We then apply it to obtain the optimal boundary regularity of harmonic maps from $RCD(K, N)$ metric measure spaces into $CAT(0)$ metric spaces. Our result is new even for harmonic functions on Lipschitz domains of Euclidean spaces. |
| title | Optimal boundary regularity of harmonic maps from $RCD(K,N)$-spaces to $CAT(0)$-spaces |
| topic | Differential Geometry Metric Geometry 58E20, 53C23 |
| url | https://arxiv.org/abs/2505.02442 |