A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón

2017/08/16 by Adrian Nachman, Nachman, Adrian I., Idan Regev +3 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1708.04759

openalex publication_date 2017/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in L² for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension 2, for conductivities σ>0 with log σ∈ H¹. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on L²-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

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