Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds
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| Format: | Preprint |
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2026
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| _version_ | 1866917553365843968 |
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| author | Wang, Yu-Zhao Hao, Jian-Hua |
| author_facet | Wang, Yu-Zhao Hao, Jian-Hua |
| contents | This paper establishes gradient estimates and Liouville-type theorems for the \(Φ\)-Laplacian equation \(Δ_Φ(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = Λ_Φ(u)\) on compact Riemannian manifolds.
Using a nonlinear \(Φ\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on \(Φ\) and \(G\), which leads to Liouville theorems for global solutions.
For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for \(p\)-harmonic functions and other quasilinear equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds Wang, Yu-Zhao Hao, Jian-Hua Differential Geometry Analysis of PDEs Primary 58J35, Secondary 35K92 This paper establishes gradient estimates and Liouville-type theorems for the \(Φ\)-Laplacian equation \(Δ_Φ(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = Λ_Φ(u)\) on compact Riemannian manifolds. Using a nonlinear \(Φ\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on \(Φ\) and \(G\), which leads to Liouville theorems for global solutions. For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for \(p\)-harmonic functions and other quasilinear equations. |
| title | Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds |
| topic | Differential Geometry Analysis of PDEs Primary 58J35, Secondary 35K92 |
| url | https://arxiv.org/abs/2606.01579 |