Inverse problems for fractional equations with a minimal number of measurements

Abstract

In this paper, we study several inverse problems associated with a fractional differential equation of the following form:which is given in a bounded domain Ω ⊂ R n , n ≥ 1. For any finite N , we show that a (k) (x), k = 0, 1, . . . , N , can be uniquely determined by N + 1 different pairs of Cauchy data in Ωe := R n \Ω. If N = ∞, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely s = 1, even for the simplest case when N = 0, a fortiori N ≥ 1. The nonlocality plays a key role in establishing the uniqueness result, and we do not utilize any linearization techniques. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.

An inverse problem for semilinear equations involving the fractional Laplacian

“…Utilizing these tools, inverse problems involving fractional operators have been greatly investigated by numerous authors in recent years. We refer readers to [GRSU20,GU21,FGKU21,QU22] for some recent works involving inverse problems for fractional linear elliptic equations, and to [LL22,LO22,Li21,Lin22,LL23] for some recent works involving inverse problems for fractional semilinear elliptic equations. Besides the theoretical study of fractional models and nonlinear models, there are also motivations for such kind of inverse problems from the practical point of view, such as anomalous diffusion processes [JR15] for fractional model and the Yang-Mills-Higgs equation [CLOP22] for the nonlinearity.…”

“…We mention that in [CLL19] the authors established the uniqueness of the obstacle and the linear potential without using the full information of the DN-map. Furthermore, in [GRSU20,LL23], only finite dimensional data set are used to establish the unique determination. All these results benefit from the nonlocal property of fractional operators.…”

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

“…Without being exhaustive, we mention the works of [Isa01, CY88, COY06] devoted to the determination of semilinear terms depending only on the solution and the determination of quasilinear terms addressed in [CK18a, EPS17, FKU22]. Finally, we mention the works of [FO20, KLU18, LLLS21, LLLS20, LLST22, KU20b, KU20a, FLL23, HL23, KKU23, CFK+21, FKU22, LL22a, Lin22, KU23, LL23, LL19] devoted to similar problems for elliptic and hyperbolic equations. Moreover, in the recent works [LLLZ22, LLL21], the authors investigated simultaneous determination problems of coefficients and initial data for both parabolic and hyperbolic equations.…”