Sign-changing solutions to the Yamabe problem on manifolds with boundary
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910155338153984 |
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| author | Clapp, Mónica Pellacci, Benedetta Pistoia, Angela |
| author_facet | Clapp, Mónica Pellacci, Benedetta Pistoia, Angela |
| contents | Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open.
In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10553 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sign-changing solutions to the Yamabe problem on manifolds with boundary Clapp, Mónica Pellacci, Benedetta Pistoia, Angela Differential Geometry Analysis of PDEs Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates. |
| title | Sign-changing solutions to the Yamabe problem on manifolds with boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2511.10553 |