The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912673467203584 |
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| author | Zhang, Jiaogen |
| author_facet | Zhang, Jiaogen |
| contents | In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-Λu \quad \textrm{in} \, Ω, \qquad u=0 \quad \textrm{on} \, \partial Ω. \] These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $Γ$-admissible eigenfunction on the smooth, strictly $Γ$-convex domain $Ω\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(Ω) \cap C^{1,1}(\overlineΩ)$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16748 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Dirichlet eigenvalue problems for some concave elliptic Hessian operators Zhang, Jiaogen Analysis of PDEs Differential Geometry In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-Λu \quad \textrm{in} \, Ω, \qquad u=0 \quad \textrm{on} \, \partial Ω. \] These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $Γ$-admissible eigenfunction on the smooth, strictly $Γ$-convex domain $Ω\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(Ω) \cap C^{1,1}(\overlineΩ)$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established. |
| title | The Dirichlet eigenvalue problems for some concave elliptic Hessian operators |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2510.16748 |