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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.21036 |
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| _version_ | 1866912787958071296 |
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| author | Quach, Van-Chuong Nguyen, Thanh-Nhan Tran, Minh-Phuong |
| author_facet | Quach, Van-Chuong Nguyen, Thanh-Nhan Tran, Minh-Phuong |
| contents | This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is strictly weaker than the VMO condition. In the complex setting, the well-posedness of this problem was recently investigated in [W. Kim, M. Vestberg, Existence, uniqueness and regularity for elliptic $p$-Laplace systems with complex coefficients,arXiv:2503.18932], where the authors established a strong accretivity condition on the leading coefficients, and this structural condition allows them to derive Schauder-type estimates for weak solutions. In our study, it has already been observed that gaining existence and uniqueness of weak solutions is possible under a natural and less restrictive assumption on the complex-valued coefficients. Following this direction, we prove a global Caderón-Zygmund-type estimate for weak solutions, from which the Morrey-space regularity follows as a consequence. This paper is a contribution to the better understanding of solution behavior and may be viewed as part of a series of works aimed at extending regularity theory in the complex-valued setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Calderón-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients Quach, Van-Chuong Nguyen, Thanh-Nhan Tran, Minh-Phuong Analysis of PDEs This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is strictly weaker than the VMO condition. In the complex setting, the well-posedness of this problem was recently investigated in [W. Kim, M. Vestberg, Existence, uniqueness and regularity for elliptic $p$-Laplace systems with complex coefficients,arXiv:2503.18932], where the authors established a strong accretivity condition on the leading coefficients, and this structural condition allows them to derive Schauder-type estimates for weak solutions. In our study, it has already been observed that gaining existence and uniqueness of weak solutions is possible under a natural and less restrictive assumption on the complex-valued coefficients. Following this direction, we prove a global Caderón-Zygmund-type estimate for weak solutions, from which the Morrey-space regularity follows as a consequence. This paper is a contribution to the better understanding of solution behavior and may be viewed as part of a series of works aimed at extending regularity theory in the complex-valued setting. |
| title | Calderón-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.21036 |