Prescribed nonlinearity helps in an anisotropic Calderón-type problem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929393923784704 |
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| author | Cârstea, Cătălin I. |
| author_facet | Cârstea, Cătălin I. |
| contents | In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(γ\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $γ$ is a smooth, matrix valued, function with a uniform lower bound. I show that boundary Dirichlet and Neumann data for this equation, in the form of a Dirichlet-to-Neumann map, determine the coefficient matrix uniquely, in dimension 3 and higher. This stands in contrast to the classical linear anisotropic Calderón problem where there is a known obstruction to uniqueness due to the invariance of the boundary data under transformations of the equation via any boundary fixing diffeomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14970 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prescribed nonlinearity helps in an anisotropic Calderón-type problem Cârstea, Cătălin I. Analysis of PDEs In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(γ\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $γ$ is a smooth, matrix valued, function with a uniform lower bound. I show that boundary Dirichlet and Neumann data for this equation, in the form of a Dirichlet-to-Neumann map, determine the coefficient matrix uniquely, in dimension 3 and higher. This stands in contrast to the classical linear anisotropic Calderón problem where there is a known obstruction to uniqueness due to the invariance of the boundary data under transformations of the equation via any boundary fixing diffeomorphism. |
| title | Prescribed nonlinearity helps in an anisotropic Calderón-type problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.14970 |