Combining the Runge approximation and the Whitney embedding theorem in\n hybrid imaging

2019/04/03 by Giovanni S. Alberti, Alberti, Giovanni S., Yves Capdeboscq +1
Computer Science · Engineering · Mathematics · #35B38 #35J25 #35R30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1904.02081

openalex publication_date 2019/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses enforcing non-vanishing constraints for solutions to a\nsecond order elliptic partial differential equation by appropriate choices of\nboundary conditions. We show that, in dimension d\≥2, under suitable\nregularity assumptions, the family of 2d solutions such that their Jacobian\nhas maximal rank in the domain is both open and dense. The case of less regular\ncoefficients is also addressed, together with other constraints, which are\nrelevant for applications to recent hybrid imaging modalities. Our approach is\nbased on the combination of the Runge approximation property and the Whitney\nprojection argument [Greene and Wu, Ann. Inst. Fourier (Grenoble), 25(1,\nvii):215-235, 1975]. The method is very general, and can be used in other\nsettings.\n

Citations

• Estimates for the dilatation of σ-harmonic mappings
• A Global Uniqueness Theorem for an Inverse Boundary Value Problem

Related

• Boundary control of elliptic solutions to enforce local constraints
• Hybrid inverse problems and redundant systems of partial differential equations
• From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations
• Volume Estimates for Singular sets and Critical Sets of Elliptic Equations with Hölder Coefficients
• On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results