2022/06/06 by Huaian Diao, Xiaoxu Fei, Diao, Huaian +5 · 4 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2206.02500
openalex publication_date 2022/06/06 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form -Δu+a(\mathbf x, u)=0, where a(\mathbf x, u) is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of f(\mathbf x, u) is contained in a bounded domain D in the sense that outside D, a(\mathbf x, u)=λu with λ∈ℂ. We establish novel unique identifiability results in several general scenarios of practical interest. These include determining the support of the inclusion (i.e. D) independent of its content (i.e. a(x, u) in D) by a single boundary measurement; and determining both D and a(x, u)|D by M boundary measurements, where M∈ℕ signifies the number of unknown coefficients in a(\mathbf x, u). The mathematical argument is based on microlocally characterising the singularities in the solution u induced by the geometric singularities of D, and does not rely on any linearisation technique.