Numerical solution of the two-dimensional Calderon problem for domains close to a disk
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910016620986368 |
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| author | Sharafutdinov, Vladimir A. Storozhuk, Konstantin V. |
| author_facet | Sharafutdinov, Vladimir A. Storozhuk, Konstantin V. |
| contents | For a compact Riemannian surface $(M,g)$ with non-empty boundary $Γ$, the Dirichlet-to-Neumann operator (DtN-map) $Λ_g:C^\infty(Γ)\to C^\infty(Γ)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_Γ$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_Γ=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $Λ_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}=\{(x,y)\mid x^2+y^2\le1\}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Numerical solution of the two-dimensional Calderon problem for domains close to a disk Sharafutdinov, Vladimir A. Storozhuk, Konstantin V. Differential Geometry 53-08, 53A45 For a compact Riemannian surface $(M,g)$ with non-empty boundary $Γ$, the Dirichlet-to-Neumann operator (DtN-map) $Λ_g:C^\infty(Γ)\to C^\infty(Γ)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_Γ$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_Γ=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $Λ_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}=\{(x,y)\mid x^2+y^2\le1\}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain. |
| title | Numerical solution of the two-dimensional Calderon problem for domains close to a disk |
| topic | Differential Geometry 53-08, 53A45 |
| url | https://arxiv.org/abs/2602.08662 |