Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913812315111424 |
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| author | Balci, Anna Sil, Swarnendu Surnachev, Mikhail |
| author_facet | Balci, Anna Sil, Swarnendu Surnachev, Mikhail |
| contents | The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order systems like the div-curl systems, Hodge-Dirac systems, and Bogovskii-type problems in the context of variable exponent spaces. Our approach yields both existence theorems and elliptic estimates. These estimates provide key results such as the Hodge decomposition theorem, Gaffney inequality, and gauge fixing. These findings are crucial for advancing the nonlinear theory related to problems involving differential forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces Balci, Anna Sil, Swarnendu Surnachev, Mikhail Analysis of PDEs 58J10, 58J32, 58A14, 35J50, 35N25 The objective of this work is to establish a systematic study of boundary value problems within the framework of differential forms and variable exponent spaces. Specifically, we investigate the Hodge Laplacian and related first order systems like the div-curl systems, Hodge-Dirac systems, and Bogovskii-type problems in the context of variable exponent spaces. Our approach yields both existence theorems and elliptic estimates. These estimates provide key results such as the Hodge decomposition theorem, Gaffney inequality, and gauge fixing. These findings are crucial for advancing the nonlinear theory related to problems involving differential forms. |
| title | Hodge Decomposition and Potentials in Variable Exponent Lebesgue and Sobolev Spaces |
| topic | Analysis of PDEs 58J10, 58J32, 58A14, 35J50, 35N25 |
| url | https://arxiv.org/abs/2504.20772 |