An inverse source problem for a fully nonlinear elliptic equation
Abstract
We study an inverse source problem for fully nonlinear elliptic equations of the form F(D2u)=f in Ω. The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities F with injective differential DF, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge--Ampère type and related examples.
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BibTeX
@misc{lin2026inverse,
title = {{An inverse source problem for a fully nonlinear elliptic equation}},
author = {Ching-Lung Lin and Yi-Hsuan Lin and Jenn-Nan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.06431},
eprint = {2606.06431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06431},
}
RIS
TY - UNPB
TI - An inverse source problem for a fully nonlinear elliptic equation
AU - Ching-Lung Lin
AU - Yi-Hsuan Lin
AU - Jenn-Nan Wang
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2606.06431
UR - https://arxiv.org/abs/2606.06431
AB - We study an inverse source problem for fully nonlinear elliptic equations of the form F(D^2u)=f in Ω. The question is whether the source term can be recovered from the Dirichlet-to-Neumann map. In two dimensions, the first linearization does not immediately give uniqueness: it leaves a natural conformal ambiguity in the linearized coefficients. For homogeneous nonlinearities F with injective differential DF, we show that this ambiguity has a precise meaning at the level of the equation itself, namely that the source is determined up to an explicit scalar factor. The main point of the paper is to show how this remaining factor can be removed. We use the second linearization to extract information which is invisible at first order, and combine it with an algebraic nondegeneracy condition on the nonlinearity. Under this condition, the residual ambiguity is forced to be trivial, and the Dirichlet-to-Neumann map uniquely determines the source. The result applies, in particular, to homogeneous admissible Hessian equations of Monge--Ampère type and related examples.
ER -