Free Access
Issue
ESAIM: COCV
Volume 26, 2020
Article Number 20
Number of page(s) 33
DOI https://doi.org/10.1051/cocv/2020005
Published online 19 February 2020
  1. G. Acosta, F.M. Bersetche and J.P. Borthagaray, A short fe implementation for a 2d homogeneous dirichlet problem of a fractional laplacian. Comput. Math. Appl. 74 (2017) 784–816. [Google Scholar]
  2. H. Antil and S. Bartels, Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling. Comput. Methods Appl. Math. 17 (2017) 661–678. [CrossRef] [Google Scholar]
  3. H. Antil and D. Leykekhman, A brief introduction to PDE-constrained optimization, in Frontiers in PDE-constrained optimization, Vol. 163. Springer, New York (2018) 3–40. [CrossRef] [Google Scholar]
  4. H. Antil and C.N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications. SIAM J. Math. Anal. 51 (2019) 2479–2503. [CrossRef] [Google Scholar]
  5. H. Antil and M. Warma, Optimal control of the coefficient for regional fractional p-Laplace equations: Approximation and convergence. Math. Control Relat. Fields. 9 (2019) 1–38. [CrossRef] [Google Scholar]
  6. H. Antil and M. Warma, Optimal control of fractional semilinear PDEs. ESAIM Control Optim. Calc. Var. 26 (2020) 30. [CrossRef] [Google Scholar]
  7. H. Antil, J. Pfefferer and M. Warma, A note on semilinear fractional elliptic equation: analysis and discretization. ESAIM: M2AN 51 (2017) 2049–2067. [CrossRef] [EDP Sciences] [Google Scholar]
  8. 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]
  9. 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]
  10. H. Antil, J. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization. Commun. Math. Sci. 16 (2018) 1395–1426. [Google Scholar]
  11. H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs. Inverse Probl. 35 (2019) 084003. [Google Scholar]
  12. H. Antil, Z. Di and R. Khatri, Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography. arXiv preprint arXiv:1907.09605, 2019. [Google Scholar]
  13. W. Arendt and R. Nittka, Equivalent complete norms and positivity. Arch. Math. 92 (2009) 414–427. [Google Scholar]
  14. W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, volume 96 of Monographs in Mathematics, 2nd edn., Birkhäuser/Springer Basel AG, Basel (2011). [CrossRef] [Google Scholar]
  15. H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces, MOS-SIAM Series on Optimization, 2nd edn.,Vol. 17. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2014). [CrossRef] [Google Scholar]
  16. U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, in Recent Advances in PDEs: Analysis, Numericsand Control, Springer, Berlin (2018), 233–249. [CrossRef] [Google Scholar]
  17. C. Bjorland, L. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian. Comm. Pure Appl. Math. 65 (2012) 337–380. [CrossRef] [Google Scholar]
  18. 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]
  19. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32 (2007) 1245–1260. [CrossRef] [Google Scholar]
  20. L.A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171 (2008) 425–461. [Google Scholar]
  21. 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]
  22. A. Carbotti, S. Dipierro and E. Valdinoci, Local density of solutions of time and space fractional equations. arXiv preprint arXiv:1810.08448, (2018). [Google Scholar]
  23. B. Claus and M. Warma, Realization of the fractional laplacian with nonlocal exterior conditions via forms method. arXiv preprint arXiv:1904.13312, (2019). [Google Scholar]
  24. 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]
  25. S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations. J. Geom. Anal. 29 (2016) 1428–1455. [Google Scholar]
  26. S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33 (2017) 377–416. [CrossRef] [Google Scholar]
  27. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23 (2013) 493–540. [Google Scholar]
  28. C. Geuzaine and J.-F. Remacle, Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Method Biomed. Eng. 79 (2009) 1309–1331. [Google Scholar]
  29. T. Ghosh, M. Salo and G. Uhlmann, The calder\’on problem for the fractional schr\” odinger equation. arXiv preprint arXiv:1609.09248, (2016). [Google Scholar]
  30. T. Ghosh, Y-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators. Commun. Partial Differ. Equ. 42 (2017) 1923–1961. [CrossRef] [Google Scholar]
  31. W. Gong, M. Hinze and Z. Zhou, Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic PDEs. J. Sci. Comput. 66 (2016) 941–967. [Google Scholar]
  32. 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]
  33. 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]
  34. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE constraints, Mathematical Modelling: Theory and Applications, Vol. 23 Springer, New York (2009). [Google Scholar]
  35. N.V. Krylov, On the paper: all functions are locally s-harmonic up to a small error, edited by Dipierro, Savin and Valdinoci, arXiv preprint arXiv:1810.07648, (2018). [Google Scholar]
  36. Ru-Yu Lai and Yi-Hsuan Lin, Global uniqueness for the fractional semilinear Schrödinger equation. Proc. Amer. Math. Soc. 147 (2019) 1189–1199. [CrossRef] [Google Scholar]
  37. P.A. Larkin and M. Whalen, Direct, near field acoustic testing. Technical report, SAE technical paper (1999). [Google Scholar]
  38. T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Contin. Dyn. Syst. 35 (2015) 6031–6068. [CrossRef] [Google Scholar]
  39. C. Louis-Rose and M. Warma, Approximate controllability from the exterior of space-time fractional wave equations. Appl. Math. Optim. (2018) 1–44. https://doi.org/10.1007/s00245-018-9530-9. [Google Scholar]
  40. 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]
  41. R. Nittka, Inhomogeneous parabolic Neumann problems. Czechoslovak Math. J. 64 (2014) 703–742. [CrossRef] [Google Scholar]
  42. X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50 (2014) 723–750. [Google Scholar]
  43. A. Rüland and M. Salo, The fractional Calderón problem: low regularity and stability. arXiv preprint arXiv:1708.06294, (2017). [Google Scholar]
  44. 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]
  45. M.I. Višik and G.I. Èskin, Convolution equations in a bounded region. Uspehi Mat. Nauk. 20 (1965) 89–152. [Google Scholar]
  46. M. Warma, A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Commun. Pure Appl. Anal. 14 (2015) 2043–2067. [CrossRef] [Google Scholar]
  47. 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]
  48. M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations. SIAM J. Control Optim. 57 (2019) 2037–2063. [Google Scholar]
  49. C.J. Weiss, B.G. van Bloemen Waanders and H. Antil, Fractional operators applied to geophysical electromagnetics. Geophys. J. Int. 220 (2020) 1242–1259. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.