Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation
Abstract
We prove uniqueness for an inverse boundary problem for the prescribed Gaussian curvature equation det D2u=K(x)(1+|∇ u|2)2 on a smooth bounded simply connected planar domain. We compare smooth positive curvatures on a common open class of Dirichlet values for which the convex problems are well posed. Equality of the nonlinear Dirichlet-to-Neumann maps, together with the prescribed first boundary jet of the curvature, implies equality of the curvatures throughout the domain. The first linearized equation has conductivity Σ=K(D2u)−1. The planar anisotropic conductivity theorem determines this tensor up to a boundary fixing diffeomorphism. We prove that the Hessian and gradient structure forces this diffeomorphism to be the identity. The difference between the pulled-back background gradient and the original gradient satisfies a local elliptic system whose components have the same scalar principal part and zero boundary Cauchy data. A first-order Carleman estimate gives unique continuation for this system. The argument uses the nonlinear boundary response at one admissible Dirichlet value and its first derivative there.
Related
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BibTeX
@misc{lin2026gauge,
title = {{Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation}},
author = {Yi-Hsuan Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04657},
eprint = {2607.04657},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04657},
}
RIS
TY - UNPB
TI - Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation
AU - Yi-Hsuan Lin
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2607.04657
UR - https://arxiv.org/abs/2607.04657
AB - We prove uniqueness for an inverse boundary problem for the prescribed Gaussian curvature equation det D^2u=K(x)(1+|∇ u|^2)^2 on a smooth bounded simply connected planar domain. We compare smooth positive curvatures on a common open class of Dirichlet values for which the convex problems are well posed. Equality of the nonlinear Dirichlet-to-Neumann maps, together with the prescribed first boundary jet of the curvature, implies equality of the curvatures throughout the domain. The first linearized equation has conductivity Σ=K(D^2u)^−1. The planar anisotropic conductivity theorem determines this tensor up to a boundary fixing diffeomorphism. We prove that the Hessian and gradient structure forces this diffeomorphism to be the identity. The difference between the pulled-back background gradient and the original gradient satisfies a local elliptic system whose components have the same scalar principal part and zero boundary Cauchy data. A first-order Carleman estimate gives unique continuation for this system. The argument uses the nonlinear boundary response at one admissible Dirichlet value and its first derivative there.
ER -