Fractional Calderón problem with L^∞-conductivities

Abstract

We study the inverse problem for the fractional conductivity equation with uniformly elliptic conductivities γ∈ L^∞(ℝⁿ). For every n≥1 and 0<s<1, we prove that the local exterior Dirichlet-to-Neumann map on any nonempty open subset of the exterior of a bounded open domain determines γ uniquely almost everywhere in ℝⁿ. No continuity, Sobolev regularity, or prescribed behavior at infinity is required. The proof works directly with the conductivity bilinear form and avoids the fractional Liouville reduction. It combines exterior determination at Lebesgue points, equality of weighted solutions, and a unique continuation principle for the fractional Laplacian in the weighted class L_s={ g∈ L¹_loc(ℝⁿ): ∫_ℝⁿ |g(x)|/(1+|x|^n+2s) dx<∞ }. Specifically, if g∈ L_s and both g and (−Δ)^s g vanish in a nonempty open set, with the latter equality understood distributionally, then g=0 almost everywhere. This result requires neither local boundedness nor Sobolev regularity of g. Its proof uses the Caffarelli--Silvestre extension and boundary unique continuation for the associated degenerate elliptic equation.

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BibTeX RIS

BibTeX

@misc{uhlmann2026fractional,
  title = {{Fractional Calderón problem with $L^\infty$-conductivities}},
  author = {Gunther Uhlmann and Philipp Zimmermann},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17524},
  eprint = {2610.17524},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17524},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Fractional Calderón problem with L^∞-conductivities
AU  - Gunther Uhlmann
AU  - Philipp Zimmermann
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17524
UR  - https://arxiv.org/abs/2610.17524
AB  - We study the inverse problem for the fractional conductivity equation with uniformly elliptic conductivities γ∈ L^∞(ℝ^n). For every n≥1 and 0<s<1, we prove that the local exterior Dirichlet-to-Neumann map on any nonempty open subset of the exterior of a bounded open domain determines γ uniquely almost everywhere in ℝ^n. No continuity, Sobolev regularity, or prescribed behavior at infinity is required. The proof works directly with the conductivity bilinear form and avoids the fractional Liouville reduction. It combines exterior determination at Lebesgue points, equality of weighted solutions, and a unique continuation principle for the fractional Laplacian in the weighted class L_s={ g∈ L^1_loc(ℝ^n): ∫_ℝ^n |g(x)|/(1+|x|^n+2s) dx<∞ }. Specifically, if g∈ L_s and both g and (−Δ)^s g vanish in a nonempty open set, with the latter equality understood distributionally, then g=0 almost everywhere. This result requires neither local boundedness nor Sobolev regularity of g. Its proof uses the Caffarelli--Silvestre extension and boundary unique continuation for the associated degenerate elliptic equation.
ER  -

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