The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients

• Yi-Hsuan Lin ^[2] ; Jesse Railo ^[3] ; Philipp Zimmermann ^[1]
  1. [1] Universitat de Barcelona

     Universitat de Barcelona

     Barcelona, España

     
  2. [2] National Yang Ming Chiao Tung University, Hsinchu, Taiwan
  3. [3] University of Cambridge, Cambridge, UK; Lappeenranta-Lahti University of Technology LUT, Lappeenranta, Finland

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 41, Nº 3, 2025, págs. 1129-1172
• Idioma: inglés
• DOI: 10.4171/RMI/1539
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• Resumen

  • We investigate the Calderón problem for a nonlocal diffusion equation depending on a globally unknown isotropic coefficient γ(x,t). The forward problem is posed on Ω×(0,T) for a domain Ω that is bounded in one direction. We first show that the Dirichlet-to-Neumann map Λγ determines γ in the measurement set. By studying various properties of the related nonlocal Neumann derivatives Nγ, we prove that both quantities ⟨Λγf,g⟩ and ⟨Nγf,g⟩ carry the same information as long as f,g:Rn∖Ω→R have disjoint supports and γ is known in supp(f)∪supp(g). We obtain the desired global uniqueness theorem using a suitable integral identity for Nγ and the Runge approximation property. The results hold for any spatial dimension n≥1. In conclusion, the main observations of this article are twofold: (1) the information of Λγ is needed for exterior determination for γ, (2) the knowledge of Nγ and γ in the measurement set is enough to recover γ in the interior.