Free Access
Issue
ESAIM: COCV
Volume 26, 2020
Article Number 42
Number of page(s) 34
DOI https://doi.org/10.1051/cocv/2019028
Published online 17 July 2020
  1. H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal pdes. Inverse Problems, 2019. [Google Scholar]
  2. H. Antil, R.H. Nochetto and P. Venegas, Controlling the Kelvin force: basic strategies and applications to magnetic drug targeting. Optim. Eng. 19 (2018) 559–589. [CrossRef] [Google Scholar]
  3. H. Antil, R.H. Nochetto and P. Venegas, Optimizing the Kelvin force in a moving target subdomain. Math. Models Methods Appl. Sci. 28 (2018) 95–130. [Google Scholar]
  4. W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Vol. 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2nd edn. (2011). [Google Scholar]
  5. W. Arendt, A.F.M. ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Comm. Partial Differ. Equ. 43 (2018) 1–24. [CrossRef] [Google Scholar]
  6. U. Biccari, Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator. Preprint arXiv:1411.7800 (2014). [Google Scholar]
  7. U. Biccari and V. Hernández-Santamaria, Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. IMA J. Math. Control Inf. 36 (2019) 1199–1235. [CrossRef] [Google Scholar]
  8. U. Biccari and V. Hernández-Santamaría, The Poisson equation from non-local to local. Electr. J. Differ. Equ. 13 (2018) 145. [Google Scholar]
  9. U. Biccari, M. Warma and E. Zuazua, Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17 (2017) 837–839. [CrossRef] [Google Scholar]
  10. U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17 (2017) 387–409. [Google Scholar]
  11. K. Bogdan, K. Burdzy and Z-Q. Chen, Censored stable processes. Probab. Theory Related Fields 127 (2003) 89–152. [CrossRef] [MathSciNet] [Google Scholar]
  12. L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian. Discrete Contin. Dyn. Syst. 36 (2016) 1813–1845. [CrossRef] [Google Scholar]
  13. L.A. Caffarelli, J-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010) 1151–1179. [CrossRef] [MathSciNet] [Google Scholar]
  14. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. [CrossRef] [MathSciNet] [Google Scholar]
  15. S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33 (2017) 377–416. [CrossRef] [Google Scholar]
  16. A.A. Dubkov, B. Spagnolo and V.V. Uchaikin, Lévy flight superdiffusion: an introduction. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008) 2649–2672. [CrossRef] [Google Scholar]
  17. H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. [MathSciNet] [Google Scholar]
  18. C.G. Gal and M. Warma, Bounded solutions for nonlocal boundary value problems on Lipschitz manifolds with boundary. Adv. Nonlinear Stud. 16 (2016) 529–550. [Google Scholar]
  19. C.G. Gal and M. Warma, Fractional in time semilinear parabolic equations and applications. In Vol. 84 of Mathematiques et Applications. Springer (2020). [Google Scholar]
  20. C.G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces. Commun. Partial Differ. Equ. 42 (2017) 579–625. [CrossRef] [Google Scholar]
  21. C.G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ. 14 (2017) 47–77. [CrossRef] [Google Scholar]
  22. T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation. Preprint arXiv:1609.09248 (2016). [Google Scholar]
  23. R. Gorenflo, F. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 34 (2007) 87–103. [Google Scholar]
  24. G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators. Adv. Math. 268 (2015) 478–528. [CrossRef] [Google Scholar]
  25. R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces. J. Differ. Equ. 254 (2013) 3352–3368. [Google Scholar]
  26. V. Keyantuo and M. Warma, On the interior approximate controllability for fractional wave equations. Discrete Contin. Dyn. Syst. 36 (2016) 3719–3739. [CrossRef] [Google Scholar]
  27. P.A. Larkin and M. Whalen, Direct, near field acoustic testing. Technical report, SAE technical paper (1999). [Google Scholar]
  28. C. Louis-Rose and M. Warma, Approximate controllability from the exterior of space-time fractional wave equations. To appear in: Appl Math Optim (2018). https://doi.org/10.1007/s00245-018-9530-9 [PubMed] [Google Scholar]
  29. Q. Lü and E. Zuazua, On the lack of controllability of fractional in time ODE and PDE. Math. Control Signals Systems 28 (2016) Art. 10, 21. [CrossRef] [Google Scholar]
  30. A.S. Lübbe, C. Bergemann, H. Riess, F. Schriever, P. Reichardt, K. Possinger, M. Matthias, B. Dörken, F. Herrmann, R. Gürtler et al., Clinical experiences with magnetic drug targeting: a phase i study with 4’-epidoxorubicin in 14 patients with advanced solid tumors. Cancer Res. 56 (1996) 4686–4693. [Google Scholar]
  31. F. Mainardi, An introduction to mathematical models, in Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London (2010). [Google Scholar]
  32. B.B. Mandelbrot and J.W. Van Ness Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  33. P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684. [Google Scholar]
  34. E. Niedermeyer and F.H.L. da Silva, Electroencephalography: basic principles, clinical applications, and related fields. Lippincott Williams & Wilkins (2005). [Google Scholar]
  35. X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101 (2014) 275–302. [Google Scholar]
  36. X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50 (2014) 723–750. [Google Scholar]
  37. X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213 (2014) 587–628. [Google Scholar]
  38. L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat 5 (2007) 79–84. [Google Scholar]
  39. W.R. Schneider, Grey noise. In Stochastic processes, physics and geometry (Ascona and Locarno, 1988). World Sci. Publ., Teaneck, NJ (1990) 676–681. [Google Scholar]
  40. R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A 144 (2014) 831–855. [CrossRef] [Google Scholar]
  41. M. Unsworth, New developments in conventional hydrocarbon exploration with electromagnetic methods. CSEG Recorder 30 (2005) 34–38. [Google Scholar]
  42. E. Valdinoci, From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49 (2009) 33–44. [Google Scholar]
  43. M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42 (2015) 499–547. [CrossRef] [Google Scholar]
  44. M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 23 (2016) 1. [CrossRef] [Google Scholar]
  45. M. Warma, On the approximate controllability from the boundary for fractional wave equations. Appl. Anal. 96 (2017) 2291–2315. [Google Scholar]
  46. M. Warma, Approximate controllabilty from the exterior of space-time fractional diffusive equations. SIAM J. Control Optim. 57 (2019) 2037–2063. [Google Scholar]
  47. M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation. Preprint arXiv:1811.10477 (2018). [Google Scholar]
  48. C.J. Weiss, B.G. Waanders and H. Antil, Fractional operators applied to geophysical electromagnetics. Preprint arXiv:1902.05096 (2019). [Google Scholar]
  49. R.L. Williams, I. Karacan and C.J. Hursch, Electroencephalography (EEG) of human sleep: clinical applications. John Wiley & Sons (1974). [Google Scholar]
  50. P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (2006) 87–99. [Google Scholar]
  51. E. Zuazua, Controllability of partial differential equations. 3ème cycle. Castro Urdiales, Espagne (2006). [Google Scholar]

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