Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911646891376640 |
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| author | Butaev, Almaz Luo, Liangbing Shanmugalingam, Nageswari |
| author_facet | Butaev, Almaz Luo, Liangbing Shanmugalingam, Nageswari |
| contents | Given a complete doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we approximate harmonic functions on a bounded domain $Ω$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(Ω)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(Ω)$ is obtained as a $Γ$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03332 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs Butaev, Almaz Luo, Liangbing Shanmugalingam, Nageswari Analysis of PDEs Metric Geometry 31E05, 46E36, 49Q20, 65Z99, 30L15 Given a complete doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we approximate harmonic functions on a bounded domain $Ω$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(Ω)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(Ω)$ is obtained as a $Γ$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs. |
| title | Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs |
| topic | Analysis of PDEs Metric Geometry 31E05, 46E36, 49Q20, 65Z99, 30L15 |
| url | https://arxiv.org/abs/2605.03332 |