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In this chapter, we study inverse problems for more general variable coefficients nonlocal operators, including unique determination of lower order coefficients, inverse obstacle problems, and leading coefficients recovering problems. More precisely, let us consider more general inverse problems variable coefficients nonlocal elliptic operators \( \left ( \mathcal {L}^s +q \right ) u =0\) in \(\Omega \) , where \(\mathcal {L}=(-\nabla \cdot \sigma \nabla )^s\) . As \(\sigma =\mathrm {I}_n\) (an \(n\times n\) identity matrix), \(\mathcal {L}^s=(-\varDelta )^s\) is the fractional Laplacian of order \(s\in (0,1)\) . We aim to study inverse coefficient problems for the identifications of coefficients q or \(\sigma \) , and related inverse obstacle problems to determine both obstacle and q will be addressed.

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Inverse Problems for Variable Coefficients Nonlocal Equations

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  • 发表日期：

    2025年

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  • Yi-Hsuan Lin,
  • Hongyu Liu

摘要

In this chapter, we study inverse problems for more general variable coefficients nonlocal operators, including unique determination of lower order coefficients, inverse obstacle problems, and leading coefficients recovering problems. More precisely, let us consider more general inverse problems variable coefficients nonlocal elliptic operators \( \left ( \mathcal {L}^s +q \right ) u =0\) in \(\Omega \) , where \(\mathcal {L}=(-\nabla \cdot \sigma \nabla )^s\) . As \(\sigma =\mathrm {I}_n\) (an \(n\times n\) identity matrix), \(\mathcal {L}^s=(-\varDelta )^s\) is the fractional Laplacian of order \(s\in (0,1)\) . We aim to study inverse coefficient problems for the identifications of coefficients q or \(\sigma \) , and related inverse obstacle problems to determine both obstacle and q will be addressed.