Inverse problems for quasi-linear elliptic systems modeling electrolysers
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arXiv
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| Format: | Preprint |
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2026
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| author | Alberti, Giovanni S. Gerner, Wadim Santacesaria, Matteo |
| author_facet | Alberti, Giovanni S. Gerner, Wadim Santacesaria, Matteo |
| contents | We investigate the electrochemical processes within an electrolyser cell, which are modelled by a coupled system of second-order quasi-linear elliptic PDEs. In this context, we study an inverse problem aiming to reconstruct both the non-linear diffusion coefficients and the phenomenological relation defining the electric potential. Our main results state that boundary measurements alone are not enough to reconstruct these non-linear quantities. However, we show that a combination of boundary and interior measurements allow for their unique reconstruction. To achieve this result we generalise a linearisation result in the context of the scalar quasi-linear Calderón problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calderón case, the generalised linearisation does not "freeze" the coefficients. We show that interior measurements are precisely what is required to achieve this freezing and thus enable the unique reconstruction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse problems for quasi-linear elliptic systems modeling electrolysers Alberti, Giovanni S. Gerner, Wadim Santacesaria, Matteo Analysis of PDEs 35J57, 35R30, 35Q92, 70F17, 92E99 We investigate the electrochemical processes within an electrolyser cell, which are modelled by a coupled system of second-order quasi-linear elliptic PDEs. In this context, we study an inverse problem aiming to reconstruct both the non-linear diffusion coefficients and the phenomenological relation defining the electric potential. Our main results state that boundary measurements alone are not enough to reconstruct these non-linear quantities. However, we show that a combination of boundary and interior measurements allow for their unique reconstruction. To achieve this result we generalise a linearisation result in the context of the scalar quasi-linear Calderón problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calderón case, the generalised linearisation does not "freeze" the coefficients. We show that interior measurements are precisely what is required to achieve this freezing and thus enable the unique reconstruction. |
| title | Inverse problems for quasi-linear elliptic systems modeling electrolysers |
| topic | Analysis of PDEs 35J57, 35R30, 35Q92, 70F17, 92E99 |
| url | https://arxiv.org/abs/2602.16906 |