Free Access
Issue
ESAIM: COCV
Volume 26, 2020
Article Number 5
Number of page(s) 30
DOI https://doi.org/10.1051/cocv/2019003
Published online 27 January 2020
  1. N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Ann. Inst. Henri Poincaré Anal. Non Linéaire 34 (2017) 439–467. [CrossRef] [Google Scholar]
  2. G. Acosta and J.P. Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55 (2017) 472–495. [Google Scholar]
  3. R.A. Adams, Sobolev spaces, in Vol. 65 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). [Google Scholar]
  4. 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]
  5. H. Antil, R. Khatri and M. Warma, External optimal control of nonlocal PDEs. Inverse Problems 35 (2019) 084003. [Google Scholar]
  6. H. Antil, R.H. Nochetto and P. Sodré, Optimal control of a free boundary problem: analysis with second-order sufficient conditions. SIAM J. Control Optim. 52 (2014) 2771–2799. [Google Scholar]
  7. 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]
  8. 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]
  9. H. Antil and C.N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications. Preprint arXiv:1803.10350 (2018). [Google Scholar]
  10. H. Antil and M. Warma, Optimal control of the coefficient for fractional p-Laplace equation: approximation and convergence. RIMS Kôkyûroku 2090 (2018) 102–116. [Google Scholar]
  11. H. Antil and M. Warma, Optimal control of the coefficient for the regional fractional p-Laplace equation: approximation and convergence. Math. Cont. Relat. Fields 9 (2019) 1–38. [CrossRef] [Google Scholar]
  12. W. Arendt, A.F.M. ter Elst and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Commun. Part. Differ. Equ. 43 (2018) 1–24. [CrossRef] [Google Scholar]
  13. H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. MOS-SIAM Series on Optimization, in Applications to PDEs and optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition (2014). [Google Scholar]
  14. U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17 (2017) 387–409. [Google Scholar]
  15. M. Biegert and M. Warma, Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad”domains. Adv. Differ. Equ. 15 (2010) 893–924. [Google Scholar]
  16. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer Science & Business Media (2013). [Google Scholar]
  17. A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization. J. Royal Soc. Interf. 11 (2014) 20140352. [CrossRef] [PubMed] [Google Scholar]
  18. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32 (2007) 1245–1260. [CrossRef] [Google Scholar]
  19. L.A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12 (2010) 1151–1179. [CrossRef] [MathSciNet] [Google Scholar]
  20. L.A. Caffarelli and P.R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33 (2016) 767–807. [CrossRef] [MathSciNet] [Google Scholar]
  21. R.S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species. Proc. Roy. Soc. Edinburgh Sect. A: Math.137 (2007) 497–518. [CrossRef] [Google Scholar]
  22. E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional. SIAM J. Optim. 22 (2012) 795–820. [Google Scholar]
  23. E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by bv-functions. Set-Valued Variat. Anal. 27 (2019) 355–379. [CrossRef] [Google Scholar]
  24. E. Casas and F. Tröltzsch, A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53 (2012) 173–206. [Google Scholar]
  25. E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [Google Scholar]
  26. W. Chen, A speculative study of 23-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. Chaos 16 (2006) 023126. [Google Scholar]
  27. F.H. Clarke, Optimization and nonsmooth analysis, in Vol. 5 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition (1990). [Google Scholar]
  28. P. Constantin, A.J. Majda and E.G Tabak, Singular front formation in a model for quasigeostrophic flow. Phys. Fluids 6 (1994) 9–11. [CrossRef] [Google Scholar]
  29. 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]
  30. M. Doman, Weak uniform rotundity of Orlicz sequence spaces. Math. Nachr. 162 (1993) 145–151. [CrossRef] [Google Scholar]
  31. A.L. Dontchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim. 31 (1995) 297–326. [Google Scholar]
  32. A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40 (2015) 235–253. [CrossRef] [Google Scholar]
  33. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, 24. Pitman, Boston, MA (1985). [Google Scholar]
  34. 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]
  35. G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems. Math. Nachr. 289 (2016) 831–844. [CrossRef] [Google Scholar]
  36. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980). [Google Scholar]
  37. J.-L. Lions and E. Magenes, in Vol. I of Non-homogeneous Boundary Value Problems and Applications. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg (1972). [Google Scholar]
  38. L. Nirenberg, Topics in Nonlinear Functional Analysis. In Vol. 6. American Mathematical Soc. (1974). [Google Scholar]
  39. J. Pfefferer, Numerical analysis for elliptic Neumann boundary control problems on polygonal domains. Ph.D. thesis, Universitätsbibliothek der Universität der Bundeswehr München (2014). [Google Scholar]
  40. M.M. Rao and Z. D. Ren, Applications of Orlicz Spaces, in Vol. 250 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (2002). [Google Scholar]
  41. X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50 (2014) 723–750. [Google Scholar]
  42. 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]
  43. P.R. Stinga and J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Commun. Part. Differ. Equ. 35 (2010) 2092–2122. [CrossRef] [Google Scholar]
  44. F. Tröltzsch, Optimal Control of Partial Differential Equations, in Vol. 112 of Graduate Studies in Mathematics. Theory, methods and applications, Translated fromthe 2005 German original by Jürgen Sprekels. American Mathematical Society, Providence, RI (2010). [Google Scholar]
  45. G.M. Viswanathan, V. Afanasyev, S.V. Buldyrev, E.J. Murphy, P.A. Prince and H.E. Stanley, Lévy flight search patterns of wandering albatrosses. Nature 381 (1996) 413. [Google Scholar]
  46. M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potent. Anal. 42 (2015) 499–547. [CrossRef] [Google Scholar]
  47. M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian. Nonlin. Differ. Equ. Appl. 23 (2016) 1. [CrossRef] [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.