Rigidity of inverse problems for nonlinear elliptic equations on manifolds
arxiv(2023)
摘要
We consider the inverse problem of determining coefficients appearing in
semilinear elliptic equations stated on Riemannian manifolds with boundary
given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a
negative answer to this problem. Owing to this obstruction, we consider a new
formulation of our inverse problem in terms of a rigidity problem. Precisely,
we consider cases where the Dirichlet-to-Neumann map of a semilinear equation
coincides with the one of a linear equation and ask whether this implies that
the equation must indeed be linear. We give a positive answer to this rigidity
problem under some assumptions imposed to the Riemannian manifold and to the
semilinear term under consideration.
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