Global uniqueness for the fractional semilinear Schrödinger equation

By Ru-Yu Lai, Yi-Hsuan Lin. Proceedings of the American Mathematical Society, 2017

59 citations recorded by the source corpus.

We study global uniqueness in an inverse problem for the fractional semilinear Schr\"{o}dinger equation $(-\Delta)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94d5b01bf09866bec2e33917af844e2ed9d27002

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Citations in the corpus

The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “Other variants of the fractional Calderón problem include semilinear fractional (magnetic) Schrödinger equation [19, 20, 24, 25], fractional heat equation [21, 35] and fractional conductivity equation [8] (see also [6, 13] for equations arising from a nonlocal Schrödinger-type elliptic operator).”
Monotonicity-based inversion of the fractional Schr\"odinger equation (2017, methodology): “Furthermore, for uniqueness results, [30] and [61] solved the Calderón problem for general nonlocal variable elliptic operators and the semilinear case, respectively.”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “y [70] in the higher order cases? (3)Is there exponential instability [68] in the higher order cases? (4)Is there uniqueness for the Calderon problem for fractional semilinear Schrodinger equations [46] in the higher order cases? (5)Do the monotonicity methods [32, 33] extend to the higher order cas”