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Geometrical optics for the fractional Helmholtz equation and applications to inverse problems

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Pith's one-line read This paper constructs geometrical optics solutions for the fractional Helmholtz equation and derives Hölder stability for recovering a potential from multi-frequency Cauchy data when the fractional order is at least 1/2.

desk verdict A genuine extension of geometrical optics to the fractional Helmholtz equation, with a real but localized regularity gap in the main stability theorem that needs fixing before the stated Hölder exponent can stand. read the letter →

arxiv 2412.14698 v1 pith:EIVZM3AZ submitted 2024-12-19 math.AP

Giovanni Covi , Maarten de Hoop , Mikko Salo This is my paper · ORCID
classification math.AP MSC 35R3035R1135S30
keywords fractionalHelmholtzequationgeometricalopticssolutionseikonaltransportHölderstabilitymulti-frequencyCauchydatageodesicraytransformCalderónproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs high-frequency approximate solutions, called geometrical optics solutions, for the fractional Helmholtz equation, and uses them to prove a stability estimate for an inverse problem. For fractional order s in [1/2,1) the solutions are essentially classical: the phase solves the usual eikonal equation |∇φ|=r and the nonlocal character of the operator appears only as an s-dependent term in the transport equation, so the wave geometry is governed by the geodesics of the metric $r^{2}$δ. For s in (0,1/2) the potential is a strong perturbation and a single-phase ansatz is impossible, forcing a hierarchy of phase functions oscillating at fractional powers of the frequency. As an application, the paper shows that the potential can be recovered from multi-frequency Cauchy data with fixed simple refraction index at Hölder rate, improving on the logarithmic stability given by the usual Runge-approximation argument for fractional wave equations. The result matters because Hölder stability is the modulus that makes numerical recovery of the coefficient feasible.

What carries the argument

The engine is a stationary-phase parametrix—an approximate high-frequency solution ansatz—for the nonlocal operator. Because the symbol |ξ|^{2s} of the fractional Laplacian is singular at ξ=0, the expansion splits the symbol into a cut-off piece and a smooth piece, applies nonstationary and stationary phase estimates, and yields the eikonal and transport equations that fix φ and the amplitudes a_l; the approximate solutions are then upgraded to exact ones by a semiclassical resolvent estimate proved with a positive-commutator argument on nontrapping domains. On the inverse side, the decisive objects are the same solutions written in polar normal coordinates of the simple metric g=$r^{2}$δ: the phase φ0=±ρ is geodesic distance, the principal amplitude is a0=b(θ)$e^{{−f+iJ(q)}}$, and substituting these into the Alessandrini identity turns the boundary-data distance δ into the geodesic ray transform I of Q. A stability estimate for the normal ray transform I*I on simple manifolds closes the argument.

What would settle it

At the end of Section 4, the estimate ‖q1−q2‖_{L2}^{2+2/t_M} ≲ $δτ^{{2s}}$+$τ^{{−α1}}$‖Q‖_{$H^{{t_M}}$} is combined with ‖Q‖_{$H^{{t_M}}$}≲1. If q1,q2 are only in H^k, the bound ‖Q‖_{$H^{{t_M}}$}≲1 holds only for t_M≤k; setting t_M=k makes the final exponent α1/(4s+2α1)·k/(k+1) explicitly k-dependent. Determining whether this k-dependence is unavoidable—for instance by constructing $H^{1}$ potentials whose stability modulus is not Hölder with any exponent independent of k—would settle the claim as stated.

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Extended reading notes

Core claim

The central discovery is that the fractional Helmholtz operator with s∈[1/2,1) admits geometrical optics solutions u=$e^{{iτφ}}$(a0+$τ^{{-α1}}$a1+...+$τ^{{-αN}}$aN)+R_N with ‖R_N‖_{H^s}=O($τ^{{-C_s N}}$), where the phase satisfies the same eikonal equation as in the classical case and the first transport equation is modified only by a zeroth-order term b_s (with the potential entering at s=1/2). For s∈(0,1/2), the same ansatz provably fails, and the paper constructs instead solutions with a sum of phase functions φ_j, showing that the potential changes the propagation of singularities. Using the former regime, the paper proves Hölder stability: ‖q1−q2‖_{L2(Ω)} ≤ C[sup_{τ≥τ0} δ(C^τ_{r,q1},C^τ_{r,q2})]^γ with γ depending only on s, obtained by converting the difference of Cauchy data, through the Alessandrini identity applied to specially chosen solutions, into a geodesic ray transform of a weighted potential difference and then optimizing over the frequency τ.

Load-bearing premise

The load-bearing premise is that the weighted coefficient Q=(q1−q2)$e^{{−2f+iJ(q1−q2)}}$$r^{{−n}}$ can be treated in arbitrarily high Sobolev norm $H^{{t_M}}$ while the theorem only assumes q1,q2∈H^k; unless the potentials are smooth or the exponent is allowed to depend on k, the advertised s-only Hölder exponent is not what the proof delivers.

Editorial extensions

If this is right

  • For s∈[1/2,1), high-frequency solutions of the fractional Helmholtz equation have a single phase satisfying the classical eikonal equation, so the wave geometry follows the geodesics of the metric r^2δ.
  • Multi-frequency Cauchy data at fixed simple refraction index determine the potential with Hölder stability, a polynomial-in-data improvement over the logarithmic modulus from fixed-frequency Runge approximation.
  • For s∈(0,1/2), no single-phase geometrical optics solution can exist; the phase must be a sum of functions oscillating at fractional powers of the frequency, reflecting the potential acting as a strong perturbation.
  • Via the fractional Liouville reduction, the same parametrix applies to the fractional conductivity equation, producing approximate solutions with amplitude prefactor γ^{-1/2}.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The regularity gap in the final interpolation step suggests the clean version of Theorem 1.3 may carry an exponent depending on the Sobolev order k of the potentials; a natural test is to let t_M=k and read off the k/(k+1) factor in the exponent.
  • If the regularity issue is resolved, the ray-transform representation of the data opens the way to explicit reconstruction of q from multi-frequency data, not just stability.
  • The multi-phase structure below s=1/2 resembles long-range scattering phenomena; the lower-order phase corrections carry extra τ-dependent information that a future inverse strategy might exploit.
  • The same multi-frequency mechanism could upgrade the fractional elasticity uniqueness result to a stability estimate if the parametrix extends to the vector-valued fractional elasticity operator the authors point to as future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs high-frequency geometrical optics (GO) approximate solutions for the fractional Helmholtz equation ((-Δ)^s - τ^{2s} r^{2s} + q)u = 0, with a detailed stationary-phase expansion that separates the regimes s∈(0,1/2) and s∈[1/2,1). In the latter regime the eikonal equation agrees with the classical one and the nonlocality affects only the zeroth-order term; in the former regime additional phase functions appear. The approximate solutions are upgraded to exact solutions by a semiclassical resolvent estimate obtained through a positive-commutator argument. The parametrix is then used to prove Theorem 1.3, a Hölder stability estimate for recovering q from multi-frequency Cauchy data when r is fixed and simple, which would improve the usual logarithmic stability arising from Runge approximation.

Significance. If Theorem 1.3 is established at the stated regularity, this is a substantial advance: it transfers classical GO and geodesic-ray-transform techniques to a nonlocal operator whose semiclassical symbol is not smooth at ξ=0, and it gives the first Hölder-type multi-frequency stability in this fractional setting. The stationary-phase expansions in Section 2 and the frequency-cutoff positive-commutator resolvent argument in Section 3 are detailed, parameter-free, and credible. A particular strength is that the parametrix remainder is controlled in semiclassical Sobolev norms with explicit dependence on the approximation order. The main load-bearing defect is in the final regularity step of Theorem 1.3, not in the GO construction itself.

major comments (2)
  1. [§4, Step 5 (end of proof of Theorem 1.3)] The final interpolation step is not justified under the hypotheses of Theorem 1.3. After the estimate involving ‖Q‖_{H^{-1}} and ‖Q‖_{H^{t_M}}, the proof states: 'Observe now that Q is independent of τ, and therefore using the high order Sobolev estimates for q1,q2 we get ‖Q‖_{H^{t_M}} ≲ 1.' But Theorem 1.3 only assumes q1,q2∈H^k, while t_M may be arbitrarily large; for fixed k there is no uniform control in H^{t_M} once t_M>k. The manuscript itself flags this gap in the very next sentence: 'However, this implies that we need to assume higher order Sobolev estimates for q1,q2.' Since t_M also enters the final Hölder exponent through t_M/(t_M+1), the stated conclusion that γ depends only on s is not established by the proof. The theorem should either assume q1,q2 are smooth with uniform C^M-bounds, or state γ=γ(s,k) and set t_M=k, with a quantitative proof of the preceding estimates in terms of k.
  2. [§4, Step 2 and Propositions 2.2, 3.1] The proof of Theorem 1.3 uses smooth-potential results without a smoothing argument, despite the theorem assuming only q1,q2∈H^k. Proposition 2.2 and Proposition 3.1 are stated for r,q∈C∞(Ω), and Step 2 invokes the Section 2 approximate solutions and Proposition 3.1 directly. In particular, the transport equations and the bounds ‖a0,j‖_{H^{t_M}} are derived for smooth q. Either Theorem 1.3 should be restated for smooth potentials with uniform bounds, or a regularization argument with quantitative control in H^k must be supplied. This is a regularity-consistency issue separate from, but related to, the final H^{t_M} gap.
minor comments (4)
  1. [Introduction, p. 1] The introduction states 'q∈C∞(Rn)', while Theorem 1.3 assumes q1,q2∈H^k; these hypotheses should be reconciled, and the regularity assumptions in the introduction should match the theorem.
  2. [Theorem 1.2] In the statement of Theorem 1.2, the transport equation is written as '2∇ϕ·∇a0 + b_s a0 = 0', but only ϕ0 has been introduced; this should be ϕ0, or ϕ should be defined explicitly.
  3. [Proposition 2.2 and Section 3] Proposition 2.2 uses the norm H^β_scl(Ω) before the semiclassical Sobolev norm is defined in Section 3; the norm should be defined or referenced before its first use.
  4. [§4, Step 5] The notation b1(θ) := I(I*IQ)ν_p·θ is not explained; the paper should state precisely how the normal ν_p is used and which stability estimate for I*I on simple manifolds is being invoked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GO parametrix is derived from the operator itself, and the stability argument relies on external ray-transform estimates; the self-flagged regularity gap is a correctness issue, not circularity.

full rationale

The paper's derivation chain is not circular. The approximate geometrical optics solutions in Theorem 1.1 and Proposition 2.2 are produced by substituting the WKB ansatz u = e^{i\tau\phi}(a_0 + \tau^{-\alpha_1}a_1 + \cdots) into the fractional Helmholtz operator and imposing cancellation of successive powers of \tau, which yields the eikonal and transport equations. No parameter is fitted to the Cauchy data, and the stability estimate in Theorem 1.3 follows from the Alessandrini identity, the constructed oscillatory solutions, and the geodesic ray transform. The self-citations to [38] and [7] are used only for standard tools: the positive commutator method, the stability estimate for I^*I on simple manifolds, and the fractional Liouville reduction. These are independent published results that do not assume the theorem being proved. The manuscript explicitly flags a limitation at the end of the proof of Theorem 1.3: 'However, this implies that we need to assume higher order Sobolev estimates for q1,q2.' That is a regularity/strength-of-hypotheses gap in the stated theorem, not a circular reduction: the proof does not define the potential difference as the Cauchy-data distance by construction, nor does any key equation reduce to an input. The central claims therefore have independent mathematical content, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on no fitted parameters. The main assumptions are standard semiclassical analysis, simple/nontrapping geometry, and an imported ray transform stability estimate. The final regularity step adds an unstated H^{t_M} requirement for Q, which is the main caveat to the theorem as stated.

assumptions (5)
  • standard math The fractional Laplacian is defined as the Fourier multiplier |ξ|^{2s}, and its high-frequency asymptotics can be computed by stationary phase away from ξ=0 with separate cutoff estimates near ξ=0.
    Used throughout Section 2, especially Step 1 of Proposition 2.2, to handle the non-smooth symbol at ξ=0.
  • standard math Semiclassical pseudodifferential calculus and positive commutator estimates apply to the operator after separating low frequencies from high frequencies.
    Used in Lemmas 3.3 and 3.4 to prove the resolvent estimate in Proposition 3.2.
  • domain assumption The geodesic X-ray transform I on a simple manifold satisfies the stability estimate ||Q||_{H^{-1}} ≤ C||I*I Q||_{L2}.
    Invoked in Step 5 of Theorem 1.3 and imported from the literature, not proved in this paper.
  • domain assumption The refraction index r is positive, r≡1 outside Ω, and (Ω,g) is simple, hence nontrapping; 0 is not a Dirichlet eigenvalue for the operators considered.
    Assumed in Theorem 1.3 and needed for the GO construction, the resolvent estimates, and the ray transform stability.
  • ad hoc to paper Q=(q1-q2)e^{-2f+iJ(q1-q2)}r^{-n} has H^{t_M} norm bounded uniformly, with t_M arbitrarily large.
    The final step of Theorem 1.3 requires arbitrarily high Sobolev regularity of Q, while the theorem only assumes q1,q2∈H^k. This is an unstated regularity assumption.

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Cite this review

Pith. "Pith review of Geometrical optics for the fractional Helmholtz equation and applications to inverse problems." pith.science (2026). https://pith.science/paper/EIVZM3AZ

@misc{pith2026241214698,
  author       = {Pith},
  title        = {Pith review of: Geometrical optics for the fractional Helmholtz equation and applications to inverse problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIVZM3AZ}},
  note         = {Machine review of arXiv:2412.14698}
}
abstract

In this paper we construct a parametrix for the fractional Helmholtz equation $((-\Delta)^s - \tau^{2s} r(x)^{2s} + q(x))u=0$ making use of geometrical optics solutions. We show that the associated eikonal equation is the same as in the classical case, while in the first transport equation the effect of nonlocality is only visible in the zero-th order term, which depends on $s$. Moreover, we show that the approximate geometrical optics solutions present different behaviors in the regimes $s\in(0,\frac 12)$ and $s\in [\frac 12,1)$. While the latter case is quite similar to the classical one, which corresponds to $s=1$, in the former case we find that the potential is a strong perturbation, which changes the propagation of singularities. As an application, we study the inverse problem consisting in recovering the potential $q$ from Cauchy data when the refraction index $r$ is fixed and simple. Using our parametrix based on the construction of approximate geometrical optics solutions, we prove that H\"older stability holds for this problem. This is a substantial improvement over the state of the art for fractional wave equations, for which the usual Runge approximation argument can provide only logarithmic stability. Besides its mathematical novelty, this study is motivated by envisioned applications in nonlocal elasticity models emerging from the geophysical sciences.

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Reference graph

Works this paper leans on

58 extracted references · 50 canonical work pages · cited by 2 Pith papers

  1. [38]

    S. Ma, S. Sahoo, and M. Salo. The anisotropic calder´ on problem at large fixed frequency on manifolds with invertible ray transform. Journal of the London Mathematical Society, 110, 07 2022. doi: 10.1112/jlms.13006

    work page doi:10.1112/jlms.13006 2022

  2. [1]

    Askes and E

    H. Askes and E. Aifantis. Gradient elasticity in statics and dynamic s: an overview of formulations, length scale identification procedures, fi nite element implementations and new results. International Journal of Solids and Structures , 48(13):1962–1990, 2011

    work page 1962

  3. [2]

    Baers, G

    H. Baers, G. Covi, and A. R¨ uland. On instability properties of the fractional calder´ on problem.Preprint arXiv:2405.08381 , 2024

    arXiv 2024

  4. [3]

    Banerjee and S

    A. Banerjee and S. Senapati. The calder´ on problem for space- time fractional parabolic operators with variable coefficients. SIAM Journal of Mathematical Analysis, 56(4), 2024

    work page 2024

  5. [4]

    Carpinteri, P

    A. Carpinteri, P. Cornetti, and A. Sapora. A fractional calculus approach to nonlocal elasticity. The European Physical Journal Special Topics , 193(1):193– 204, 2011

    work page 2011

  6. [5]

    Ceki´ c, Y-H

    M. Ceki´ c, Y-H. Lin, and A. R¨ uland. The Calder´ on problem for t he fractional Schr¨ odinger equation with drift. Calc. Var. Partial Differential Equations , 59 (3):Paper No. 91, 46, 2020. ISSN 0944-2669. doi: 10.1007/s0052 6-020-01740-6. URL https://doi.org/10.1007/s00526-020-01740-6

    doi:10.1007/s0052 2020

  7. [6]

    Choulli and E

    M. Choulli and E. Ouhabaz. Fractional anisotropic calder´ on pro blem on com- plete riemannian manifolds. Communications in Contemporary Mathematics , 26(9), 2024

    work page 2024

  8. [7]

    G. Covi. Inverse problems for a fractional conductivity equatio n. Nonlinear Anal., 193:111418, 18, 2020. ISSN 0362-546X. doi: 10.1016/j.na.2019.0 1.008. URL https://doi.org/10.1016/j.na.2019.01.008

    work page doi:10.1016/j.na.2019.0 2020

Show all 58 references
  1. [8]

    G. Covi. An inverse problem for the fractional Schr¨ odinger equ a- tion in a magnetic field. Inverse Problems , 36(4):045004, 24,

  2. [9]

    G. Covi. Uniqueness for the fractional Calder´ on problem with qu asilocal per- turbations. Preprint, arXiv:2110.11063, to appear in SIAM Journal on Ma the- matical Analysis (SIMA) , 2021

    2021 arXiv

  3. [10]

    G. Covi. Uniqueness for the anisotropic fractional conductivit y equation. Preprint arXiv:2212.11331 , 2022

    2022 arXiv

  4. [11]

    G. Covi. The inverse problem for the fractional conductivity eq uation: a survey. Preprint arXiv:2408.14200 , 2024

    2024 arXiv

  5. [12]

    Covi and M

    G. Covi and M. Lassas. An inverse problem for fractional rand om walks on finite graphs. Preprint arXiv:2408.05072 , 2024

    2024 arXiv

  6. [13]

    G. Covi, K. M¨ onkk¨ onen, and J. Railo. Unique continuation prop erty and poincar` e inequality for higher order fractional laplacians with applic ations in inverse problems. Inverse Problems & Imaging , 15(4):641–681, 2021

    2021

  7. [14]

    G. Covi, M. A. Garcia-Ferrero, and A. R¨ uland. On the calder´ on problem for non- local schr¨ odinger equations with homogeneous, directionally antilo cal principal symbols. Journal of Differential Equations , 341:79–149, 2022

    2022

  8. [15]

    G. Covi, K. M¨ onkk¨ onen, J. Railo, and G. Uhlmann. The higher order fractional calder´ on problem for linear local operators: uniqueness. Advances in Mathemat- ics, 399, 2022

    2022

  9. [16]

    G. Covi, M. de Hoop, and M. Salo. Uniqueness in an inverse problem of fractional elasticity. Proceedings of the Royal Society A , 479(2278), 2024

    2024

  10. [17]

    G. Covi, J. Railo, T. Tyni, and P. Zimmermann. Stability estimates f or the in- verse fractional conductivity problem. SIAM Journal on Mathematical Analysis , 56(22), 2024

    2024

  11. [18]

    Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou. Analysis an d ap- proximation of nonlocal diffusion problems with volume constraints. SIAM Rev., 54(4):667–696, 2012. ISSN 0036-1445. doi: 10.1137/11083329 4. URL https://doi.org/10.1137/110833294

    2012 doi

  12. [19]

    Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou. A nonlocal vector calcu- lus, nonlocal volume-constrained problems, and nonlocal balance la ws. Math. Models Methods Appl. Sci. , 23(3):493–540, 2013. ISSN 0218-2025. doi: 10.1142/ S0218202512500546. URL https://doi.org/10.1142...

    2013 doi

  13. [20]

    A. Eringen. Linear theory of nonlocal elasticity and dispersion o f plane waves. International Journal of Engineering Science , 10(5):425–435, 1972

    1972

  14. [21]

    A. Eringen. Theory of Nonlocal Elasticity and Some Applica- tions. Defense Technical Information Center, Princeton., 1984. URL https://books.google.fi/books?id=Qt7stgAACAAJ

    1984

  15. [22]

    A. Eringen. Nonlocal continuum field theories . Springer-Verlag, New York, 2002. ISBN 0-387-95275-6

    2002

  16. [23]

    Eringen, C

    A. Eringen, C. Speziale, and B. Kim. Crack-tip problem in non-loca l elasticity. Journal of the Mechanics and Physics of Solids , 25(5):339–355, 1977. ISSN 0022-5096

    1977

  17. [24]

    Failla and M

    G. Failla and M. Zingales. Advanced materials modelling via fractiona l calculus: Challenges and perspectives, 2020

    2020

  18. [25]

    Feizmohammadi, K

    A. Feizmohammadi, K. Krupchyk, and G. Uhlmann. Calder´ on pro blem for fractional schr¨ odinger operators on closed riemannian manifolds . Preprint arXiv:2407.16866, 2024

    2024 arXiv

  19. [26]

    Ghosh, Y.-H

    T. Ghosh, Y.-H. Lin, and J. Xiao. The calder´ on problem for varia ble coefficients nonlocal elliptic operators. Communications in Partial Differential Equations , 42(12):1923–1961, 2017

    1923

  20. [27]

    Ghosh, A

    T. Ghosh, A. R¨ uland, M. Salo, and G. Uhlmann. Uniqueness and r econstruc- tion for the fractional calder´ on problem with a single measurement . Journal of Functional Analysis, 279(1), 2020. 30 GIOV ANNI COVI, MAARTEN DE HOOP, AND MIKKO SALO

    2020

  21. [28]

    Ghosh, M

    T. Ghosh, M. Salo, and G. Uhlmann. The calder´ on problem for th e fractional schr¨ odinger equation.Analysis and PDEs , 13(2):455–475, 2020

    2020

  22. [29]

    A. Griffith. The phenomena of rupture and flow in solids. Philosophical Trans- actions, Series A , 221:163–198, 1920

    1920

  23. [30]

    H¨ ormander.The analysis of linear partial differential operators

    L. H¨ ormander.The analysis of linear partial differential operators. I . Classics in Mathematics. Springer-Verlag, Berlin, 2003. ISBN 3-540-00662- 1. doi: 10.1007/ 978-3-642-61497-2. URL https://doi.org/10.1007/978-3-642-61497-2 . Distribution theory and Fourier analysis, Repr...

    2003 doi

  24. [31]

    Jin and W

    B. Jin and W. Rundell. A tutorial on inverse problems for anoma- lous diffusion processes. Inverse Problems , 31(3):035003, 40, 2015. ISSN 0266-5611. doi: 10.1088/0266-5611/31/3/035003. URL https://doi.org/10.1088/0266-5611/31/3/035003

    2015 doi

  25. [32]

    H. Koch, A. R¨ uland, and M. Salo. On instability mechanisms for inv erse prob- lems. Ars Inveniendi Analytica , 7, 2021

    2021

  26. [33]

    Kow, Y.-H

    P.-Z. Kow, Y.-H. Lin, and J.-N. Wang. The calder´ on problem for t he fractional wave equation: Uniqueness and optimal stability. SIAM Journal of Mathematical Analysis, 54(3), 2022

    2022

  27. [34]

    Lai, Y-H

    R-Y. Lai, Y-H. Lin, and A. R¨ uland. The Calder´ on problem for a space-time fractional parabolic equation. SIAM J. Math. Anal. , 52(3): 2655–2688, 2020. ISSN 0036-1410. doi: 10.1137/19M1270288. U RL https://doi.org/10.1137/19M1270288

    2020 doi

  28. [35]

    L. Li. A semilinear inverse problem for the fractional magnetic La placian. Preprint, arXiv:2005.06714 , 2020

    2005 arXiv

  29. [36]

    L. Li. The Calder´ on problem for the fractional magnetic opera tor. Inverse Problems, 36(7):075003, 14, 2020. ISSN 0266-5611. doi: 10.1088/1361-6 420/ ab8445. URL https://doi.org/10.1088/1361-6420/ab8445

    2020 doi

  30. [37]

    Lyakhovsky, Y

    V. Lyakhovsky, Y. Ben-Zion, and A. Agnon. Distributed damag e, faulting, and friction. Journal of Geophysical Research: Solid Earth , 102(B12):27635–27649, 1997

    1997

  31. [39]

    R. Mindlin. Microstructure in linear elasticity. Technical report, Columbia Univ., New York, Dept. of Civil Engineering and Engineering Mechanics , 1963

    1963

  32. [40]

    R. Mindlin. Second gradient of strain and surface-tension in linea r elasticity. International Journal of Solids and Structures , 1(4):417–438, 1965

    1965

  33. [41]

    Paternain, M

    G. Paternain, M. Salo, and G. Uhlmann. Geometric Inverse Problems : With Emphasis on Two Dimensions. Cambridge University Press. Cambridge Studies in Advanced Mathematics, 204, 2023

    2023

  34. [42]

    T. V. Petersdorff and E. P. Stephan. Decompositions in edge an d corner singu- larities for the solution of the dirichlet problem of the laplacian in a polyh edron. Mathematische Nachrichten , 149(1):71–103, 1990

    1990

  35. [43]

    Railo and P

    J. Railo and P. Zimmermann. Fractional calder´ on problems and p oincar´ e in- equalities on unbounded domains. Journal of Spectral Theory , 13:63–131, 2023

    2023

  36. [44]

    Railo and P

    J. Railo and P. Zimmermann. Counterexamples to uniqueness in th e inverse frac- tional conductivity problem with partial data. Inverse Problems and Imaging , 17(2), 2023

    2023

  37. [45]

    Railo and P

    J. Railo and P. Zimmermann. Low regularity theory for the invers e fractional conductivity problem. Nonlinear Analysis, 239, 2024

    2024

  38. [46]

    R¨ uland and M

    A. R¨ uland and M. Salo. Quantitative approximation properties for the fractional heat equation. Math. Control Relat. Fields , 10(1):1–26, 2020. ISSN 2156-8472. GEOMETRICAL OPTICS FOR THE FRACTIONAL HELMHOLTZ EQUATION 3 1 doi: 10.3934/mcrf.2019027. URL https://doi.org/10.3934/...

    2020 doi

  39. [47]

    R¨ uland

    A. R¨ uland. On single measurement stability for the fractional c alder´ on problem. SIAM Journal of Mathematical Analysis , 53(5), 2021

    2021

  40. [48]

    R¨ uland and M

    A. R¨ uland and M. Salo. Exponential instability in the fractional c alder´ on prob- lem. Inverse Problems , 34(4):045003, February 2018

    2018

  41. [49]

    R¨ uland and M

    A. R¨ uland and M. Salo. The fractional calder´ on problem: low re gularity and stability. Nonlinear Analysis, 193, 2020

    2020

  42. [50]

    R¨ uland and E

    A. R¨ uland and E. Sincich. Lipschitz stability for the finite dimensional fractional calder´ on problem with finite cauchy data.Inverse Problems and Imaging , 13(5), 2019

    2019

  43. [51]

    M. Salo. The fractional Calder´ on problem. Journ´ ees ´ equations aux d´ eriv´ ees partielles, Exp. No. 7:8p, 2017

    2017

  44. [52]

    Tarasov and E

    V. Tarasov and E. Aifantis. On fractional and fractal formula tions of gradient linear and nonlinear elasticity. Acta Mech. , 230(6):2043– 2070, 2019. ISSN 0001-5970. doi: 10.1007/s00707-019-2373-x . URL https://doi.org/10.1007/s00707-019-2373-x

    2019 doi

  45. [53]

    Valdinoci

    E. Valdinoci. From the long jump random walk to the fractional lap lacian. Bol. Soc. Esp. Mat. Apl. SeMA , 49, 2009

    2009

  46. [54]

    Wang and J

    B. Wang and J. Li. Nonlocal elastic theory for a medium with one or more rigid inclusions - mode iii deformation. European Journal of Mechanics - A/Solids , 93:104532, 2022

    2022

  47. [55]

    Wang and B

    W. Wang and B. Spencer. Does elastic stress modify the equilibriu m corner angle? Journal of the Mechanics and Physics of Solids , 167:105003, October 2022

    2022

  48. [56]

    Zorica and L

    D. Zorica and L. Oparnica. Energy dissipation for hereditary an d energy con- servation for non-local fractional wave equations. Philos. Trans. Roy. Soc. A , 378(2172):20190295, 24, 2020. ISSN 1364-503X

    2020

  49. [57]

    M. Zworski. Semiclassical Analysis. Graduate studies in mathematics. American mathematical Society, 2012. Department of Mathematics and Statistics, University of He lsinki, Finland Email address : giovanni.covi@helsinki.fi Department of Computational and Applied Mathematics, R...

    2012

  50. [2020]

    doi: 10.1088/1361-6420/ab661a

    ISSN 0266-5611. doi: 10.1088/1361-6420/ab661a. URL https://doi.org/10.1088/1361-6420/ab661a. GEOMETRICAL OPTICS FOR THE FRACTIONAL HELMHOLTZ EQUATION 2 9

    doi

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