Fioralba Cakoni – Rutgers University
Date/Time/Location
Thursday, September 11th, 2025, 12:10 pm; Hill Center 705
The Control of Scattered Field for Linear and Nonlinear Scattering Media
Many imaging methods in inverse scattering rely on the ability to superimpose scattering data so that the resulting scattered field corresponds to that of a point source. In the frequency domain, for compact inhomogeneities, this leads to solving two elliptic PDEs in a bounded region with a prescribed difference in Cauchy data. The case of zero scattered field leads to the study of the transmission eigenvalue problem and the regularity of free boundaries. In this talk, we introduce this concept for linear media and review key results on the transmission eigenvalue problem, its resolvent, and non-scattering phenomena. We then present recent results on the scattering problem for a nonlinear medium with compact support in the second-harmonic generation. When such a medium is probed with monochromatic light beams at a given frequency, it generates additional waves at the double frequency. The response of the medium is governed by a system of two coupled semilinear PDEs. We explore the possibility that the generated double frequency wave remains localized within the support of the medium, effectively rendering the nonlinear interaction with the probing wave invisible to an outside observer