Nonlinear elliptic Dirichlet boundary value problems on time scales
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911439589998592 |
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| author | Bandyopadhyay, Shalmali Çetinkaya, F. Ayça Cuchta, Tom |
| author_facet | Bandyopadhyay, Shalmali Çetinkaya, F. Ayça Cuchta, Tom |
| contents | We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10335 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonlinear elliptic Dirichlet boundary value problems on time scales Bandyopadhyay, Shalmali Çetinkaya, F. Ayça Cuchta, Tom Analysis of PDEs Spectral Theory 34N05, 35J65, 35J61, 35J15 We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting. |
| title | Nonlinear elliptic Dirichlet boundary value problems on time scales |
| topic | Analysis of PDEs Spectral Theory 34N05, 35J65, 35J61, 35J15 |
| url | https://arxiv.org/abs/2602.10335 |