Some optimal control and shape optimisation problems for bulk-surface cooperative systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908381276536832 |
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| author | Gentile, Andrea Mazari-Fouquer, Idriss Prunier, Raphaël |
| author_facet | Gentile, Andrea Mazari-Fouquer, Idriss Prunier, Raphaël |
| contents | The goal of this paper is to address some optimal control and shape optimisation problems arising from bulk-surface cooperative systems. The basic model under consideration is the following: letting $Ω$ be a fixed domain, we assume that a population (with density $u$) lives inside $Ω$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial Ω$ and can access other resources $g$. These two populations are coupled in a cooperative manner by a constant exchange rate at the boundary, leading to a non-standard PDE system that has already been studied in previous works by Bogosel, Giletti and Tellini, for its connection with road-field models. Building on the considerations of the aforementioned previous works, we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain $Ω$ and on the surface $\partial Ω$, i.e. how to spread resources in order to guarantee an optimal survival of the two species. We establish rigid Talenti inequalities and comparison results when $Ω$ is a ball, extending in particular the results of J. J. Langford on symmetrisation for Neumann and Robin problems. Second, when the resources distribution $f$ and $g$ are constant, we provide a partial analysis of the natural shape optimisation problem: which shape $Ω$ maximises the survival rate of the two species? Namely, we show that in certain regimes there can be no optimal shape and, by computing second-order shape derivatives, we investigate the local optimality of the ball. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_20865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some optimal control and shape optimisation problems for bulk-surface cooperative systems Gentile, Andrea Mazari-Fouquer, Idriss Prunier, Raphaël Analysis of PDEs Optimization and Control The goal of this paper is to address some optimal control and shape optimisation problems arising from bulk-surface cooperative systems. The basic model under consideration is the following: letting $Ω$ be a fixed domain, we assume that a population (with density $u$) lives inside $Ω$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial Ω$ and can access other resources $g$. These two populations are coupled in a cooperative manner by a constant exchange rate at the boundary, leading to a non-standard PDE system that has already been studied in previous works by Bogosel, Giletti and Tellini, for its connection with road-field models. Building on the considerations of the aforementioned previous works, we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain $Ω$ and on the surface $\partial Ω$, i.e. how to spread resources in order to guarantee an optimal survival of the two species. We establish rigid Talenti inequalities and comparison results when $Ω$ is a ball, extending in particular the results of J. J. Langford on symmetrisation for Neumann and Robin problems. Second, when the resources distribution $f$ and $g$ are constant, we provide a partial analysis of the natural shape optimisation problem: which shape $Ω$ maximises the survival rate of the two species? Namely, we show that in certain regimes there can be no optimal shape and, by computing second-order shape derivatives, we investigate the local optimality of the ball. |
| title | Some optimal control and shape optimisation problems for bulk-surface cooperative systems |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2505.20865 |