Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide
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| Format: | Preprint |
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2025
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| _version_ | 1866929690449543168 |
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| author | Kumar, Mandeep Zimmermann, Philipp |
| author_facet | Kumar, Mandeep Zimmermann, Philipp |
| contents | The main goal of this article is to establish Hölder stability estimates for the Calderón problem related to a relativistic wave equation. The principal novelty of this article is that the partial differential equation (PDE) under consideration depends on three unknown potentials, namely a temporal dissipative potential $A_0$, a spatial vector potential $A$ and an external potential $Φ$. Moreover, the PDE is posed in an infinite waveguide geometry $Ω=ω\times\mathbb{R}$ and not on a bounded domain. For our proof it is essential that the potentials are time-independent as a key tool in this work are pointwise estimates for the Radon transform of the vector potential $\mathcal{A}=(A_0,\mathrm{i} A)$ and external potential $Φ$. Furthermore, the demonstrated stability estimates hold for a wide range of $H^s$ Sobolev scales and a main contribution is to explicitly determine the dependence of the involved constants and the Hölder exponent on the Sobolev exponents of the potentials $A_0,A$ and $Φ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_17308 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide Kumar, Mandeep Zimmermann, Philipp Analysis of PDEs 35R30, 35L05, 44A12 The main goal of this article is to establish Hölder stability estimates for the Calderón problem related to a relativistic wave equation. The principal novelty of this article is that the partial differential equation (PDE) under consideration depends on three unknown potentials, namely a temporal dissipative potential $A_0$, a spatial vector potential $A$ and an external potential $Φ$. Moreover, the PDE is posed in an infinite waveguide geometry $Ω=ω\times\mathbb{R}$ and not on a bounded domain. For our proof it is essential that the potentials are time-independent as a key tool in this work are pointwise estimates for the Radon transform of the vector potential $\mathcal{A}=(A_0,\mathrm{i} A)$ and external potential $Φ$. Furthermore, the demonstrated stability estimates hold for a wide range of $H^s$ Sobolev scales and a main contribution is to explicitly determine the dependence of the involved constants and the Hölder exponent on the Sobolev exponents of the potentials $A_0,A$ and $Φ$. |
| title | Hölder stability estimates for the determination of time-independent potentials in a relativistic wave equation in an infinite waveguide |
| topic | Analysis of PDEs 35R30, 35L05, 44A12 |
| url | https://arxiv.org/abs/2501.17308 |