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Variational inequalities for the spectral fractional Laplacian

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Abstract

In this paper we study obstacle problems for the Navier (spectral) fractional Laplacian (−ΔΩ)s of order s ∈ (0,1) in a bounded domain Ω ⊂ Rn.

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References

  1. R. Alvarado, D. Brigham, V. Maz’ya, M. Mitrea, and E. Ziadé, “On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf–Oleinik boundary point principle,” J. Math. Sci. 176, 281–360 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Barrios, A. Figalli, and X. Ros-Oton, “Global regularity for the free boundary in the obstacle problem for the fractional Laplacian,” Preprint (2015). arXiv:1506.04684

    Google Scholar 

  3. G. Buttazzo and G. Dal Maso, “An existence result for a class of shape optimization problems,” Arch. Rational Mech. Anal. 122 (2), 183–195 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Caffarelli and A. Figalli, “Regularity of solutions to the parabolic fractional obstacle problem,” J. Reine Angew. Math. 680, 191–233 (2013).

    MathSciNet  MATH  Google Scholar 

  5. L. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Commun. Part. Differ. Equations 32 (7–9), 1245–1260 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  6. L. 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 (2), 425–461 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, “Regularity of radial extremal solutions for some non-local semilinear equations,” Commun. Part. Differ. Equations 36 (8), 1353–1384 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Dal Maso, An Introduction to -Convergence, Progress in Nonlinear Differential Equations and Their Applications, Vol. 8 (Birkhäuser, Boston, MA, 1993).

    Google Scholar 

  9. N. Dunford and J. T. Schwartz, Linear Operators (Wiley, New York, 1988), Part II.

    MATH  Google Scholar 

  10. N. Garofalo and A. Petrosyan, “Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem,” Invent. Math. 177 (2), 415–461 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Grubb, “Regularity of spectral fractional Dirichlet and Neumann problems,” Math. Nachr. 289 (7), 831–844 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators (Birkhäuser, Basel, 2006).

    MATH  Google Scholar 

  13. A. Iannizzotto, S. Mosconi and M. Squassina, “ versus -weighted minimizers,” NoDEA Nonlinear Differ. Equations Appl. 22 (3), 477–497 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. I. Kamynin and B. N. Himcenko, “Theorems of Giraud type for second order equations with a weakly degenerate nonnegative characteristic part,” Sib. Math. J. 18, 76–91 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, reprint of the 1980 original, Classics in Applied Mathematics (SIAM, Philadelphia, PA, 2000), Vol. 31.

    MATH  Google Scholar 

  16. H. Lewy and G. Stampacchia, “On the regularity of the solution of a variational inequality,” Commun. Pure Appl. Math. 22, 153–188 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Lewy and G. Stampacchia, “On the smoothness of superharmonics which solve a minimum problem,” J. Anal. Math. 23, 227–236 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Musina and A. I. Nazarov, “On fractional Laplacians,” Commun. Part. Differ. Equations 39 (9), 1780–1790 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Musina and A. I. Nazarov, “On fractional Laplacians 2,” Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (6), 1667–1673 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Musina, A. I. Nazarov, and K. Sreenadh, “Variational inequalities for the fractional Laplacian,” Potential Anal. (2016). doi 10.1007/s11118-016-9591-9

    Google Scholar 

  21. A. Petrosyan and C. A. Pop, “Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift,” J. Funct. Anal. 268 (2), 417–472 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  22. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, Vo. 3 (de Gruyter, Berlin, 1996).

    MATH  Google Scholar 

  23. R. Servadei and E. Valdinoci, “Lewy–Stampacchia type estimates for variational inequalities driven by (non) local operators,” Rev. Mat. Iberoam. 29 (3), 1091–1126 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator,” Commun. Pure Appl. Math. 60 (1), 67–112 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  25. P. R. Stinga and J. L. Torrea, “Extension problem and Harnack’s inequality for some fractional operators,” Commun. Part. Differ. Equations 35 (11), 2092–2122 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Deutscher Verlag der Wissenschaften, Berlin, 1978).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica ed Informatica, Università di Udine, via delle Scienze, 206–33100, Udine, Italy

    R. Musina

  2. St. Petersburg Department of Steklov Institute, St. Petersburg, 190005, Russia

    A. I. Nazarov

  3. St. Petersburg State University, St. Petersburg, 190005, Russia

    A. I. Nazarov

Authors
  1. R. Musina
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  2. A. I. Nazarov
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Correspondence to R. Musina or A. I. Nazarov.

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Dedicated to the memory of S.I. Pohozaev

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Musina, R., Nazarov, A.I. Variational inequalities for the spectral fractional Laplacian. Comput. Math. and Math. Phys. 57, 373–386 (2017). https://doi.org/10.1134/S0965542517030113

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