Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909089184874496 |
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| author | Ruf, Thomas |
| author_facet | Ruf, Thomas |
| contents | This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17440 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application Ruf, Thomas Analysis of PDEs Functional Analysis This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type. |
| title | Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2401.17440 |