Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations

2022/04/25 by Liimatainen, Tony, Lin, Yi-Hsuan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.11774

Abstract

We study inverse source problems associated to semilinear elliptic equations of the form Δu(x)+a(x,u)=F(x), on a bounded domain Ω⊂ ℝⁿ, n≥ 2. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities a(x,u). This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both a(x,u) and F(x) uniquely from the associated DN map. Moreover, for general nonlinearities a(x,u), we show that we can recover the derivatives ∂_u^ka(x,u) and the source F(x) up to a gauge. Especially, we recover general polynomial nonlinearities up to a gauge and generalize results of [FO20,LLLS20] by removing the assumption that u≡ 0 is a solution.

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