TY - JOUR
T1 - Uniqueness results for inverse source problems for semilinear elliptic equations
AU - Liimatainen, Tony
AU - Lin, Yi Hsuan
N1 - Publisher Copyright:
© 2024 The Author(s). Published by IOP Publishing Ltd.
PY - 2024/4
Y1 - 2024/4
N2 - Abstract We study inverse source problems associated to semilinear elliptic equations of the form formula presented on a bounded domain Ω ⊂ R n , n ⩾ 2 . We show that it is possible to use nonlinearity to recover both the source F and the nonlinearity a ( x , u ) simultaneously and uniquely for a class of nonlinearities. This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, includes a class of polynomials, which we characterize, and exponential nonlinearities. For general nonlinearities a ( x , u ) , we recover the source F(x) and the Taylor coefficients ∂ u k a ( x , u ) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683-719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553-80) by removing the assumption that u ≡ 0 is a solution. To prove our results, we consider linearizations around possibly large solutions. Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.
AB - Abstract We study inverse source problems associated to semilinear elliptic equations of the form formula presented on a bounded domain Ω ⊂ R n , n ⩾ 2 . We show that it is possible to use nonlinearity to recover both the source F and the nonlinearity a ( x , u ) simultaneously and uniquely for a class of nonlinearities. This is in contrast to inverse source problems for linear equations, which always have a natural (gauge) symmetry that obstructs the unique recovery of the source. The class of nonlinearities for which we can uniquely recover the source and nonlinearity, includes a class of polynomials, which we characterize, and exponential nonlinearities. For general nonlinearities a ( x , u ) , we recover the source F(x) and the Taylor coefficients ∂ u k a ( x , u ) up to a gauge symmetry. We recover general polynomial nonlinearities up to the gauge symmetry. Our results also generalize results of Feizmohammadi and Oksanen (2020 J. Differ. Equ. 269 4683-719), Lassas et al (2020 Rev. Mat. Iberoam. 37 1553-80) by removing the assumption that u ≡ 0 is a solution. To prove our results, we consider linearizations around possibly large solutions. Our results can lead to new practical applications, because we show that many practical models do not have the obstruction for unique recovery that has restricted the applicability of inverse source problems for linear models.
KW - gauge invariance
KW - Gross-Pitaevskii equation
KW - inverse problems
KW - inverse source problems
KW - semilinear elliptic equations
KW - sine-Gordon equation
UR - http://www.scopus.com/inward/record.url?scp=85187962805&partnerID=8YFLogxK
U2 - 10.1088/1361-6420/ad3088
DO - 10.1088/1361-6420/ad3088
M3 - Article
AN - SCOPUS:85187962805
SN - 0266-5611
VL - 40
JO - Inverse Problems
JF - Inverse Problems
IS - 4
M1 - 045030
ER -