A Shape Design Approximation for Degenerate Partial Differential Equations and Its Application
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917459142901760 |
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| author | Guo, Bao-Zhu Yang, Dong-Hui Zhong, Jie |
| author_facet | Guo, Bao-Zhu Yang, Dong-Hui Zhong, Jie |
| contents | In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02783 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Shape Design Approximation for Degenerate Partial Differential Equations and Its Application Guo, Bao-Zhu Yang, Dong-Hui Zhong, Jie Analysis of PDEs Optimization and Control In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations. |
| title | A Shape Design Approximation for Degenerate Partial Differential Equations and Its Application |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2605.02783 |