On the spectrum of two different fractional operators

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Volumn 144, Issue 4, 2014, Pages 831-855

  Servadei, Raffaella ^a   Valdinoci, Enrico ^b,c,d  

^a UNIVERSITY OF CALABRIA   (Italy)

^b LEIBNIZ INSTITUT FÜR KATALYSE E V AN DER UNIVERSITÄT ROSTOCK   (Germany)

^c UNIVERSITY OF MILAN   (Italy)

^d CNR   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84904974819     PISSN: 03082105     EISSN: 14737124     Source Type: Journal    
DOI: 10.1017/S0308210512001783     Document Type: Article

References (39)

1. F. J. Almgren and E. H. Lieb. Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2 (1989), 683-773
2. B. Barrios, E. Colorado, A. de Pablo and U. Sánchez. On some critical problems for the fractional Laplacian operator. J. Diff. Eqns 252 (2012), 6133-6162
3. W. Beckner. Sobolev inequalities, the Poisson semigroup, and analysis on the sphere on SN. Proc. Natl Acad. Sci. USA 89 (1992), 4816-4819
4. R. M. Blumenthal, R. K. Getoor and D. B. Ray. On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc. 99 (1961), 540-554
5. C. Brändle, E. Colorado, A. de Pablo and U. Sánchez. A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. A143 (2013), 39-71
6. X. Cabré and J. Solà-Morales. Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58 (2005), 1678-1732
7. X. Cabré and J. Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224 (2010), 2052-2093
8. L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Commun. PDEs 32 (2007), 1245-1260
9. L. Caffarelli and L. Silvestre. Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62 (2009), 597-638
10. L. Caffarelli and L. Silvestre. Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Analysis 200 (2011), 59-88
11. A. Capella. Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains. Commun. Pure Appl. Analysis 10 (2011), 1645-1662
12. A. Capella, J. Dávila, L. Dupaigne and Y. Sire. Regularity of radial extremal solutions for some non-local semilinear equations. Commun. PDEs 36 (2011), 1353-1384
13. R. de la Llave and E. Valdinoci. A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations. Annales Inst. H. Poincaré Analyse Non Lińeaire 26 (2009), 1309-1344
14. E. di Nezza, G. Palatucci and E. Valdinoci. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521-573
15. L. C. Evans. Partial differential equations, Graduate Studies in Mathematics, vol. 19 (Providence, RI: American Mathematical Society, 1998
16. E. B. Fabes, C. E. Kenig and R. P. Serapioni. The local regularity of solutions of degenerate elliptic equations. Commun. PDEs 7 (1982), 77-116
17. M. M. Fall. Semilinear elliptic equations for the fractional Laplacian with Hardy potential. Preprint, arXiv:1109.5530
18. A. Fiscella. Saddle point solutions for non-local elliptic operators. Preprint, arXiv: 1210.8401
19. A. Fiscella, R. Servadei and E. Valdinoci. A resonance problem for non-local elliptic operators. Z. Analysis Anwend. 32 (2013), 411-431
20. R. K. Getoor. First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101 (1961), 75-90
21. D Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order, 2nd edn (Springer, 1983
22. L. Hörmander. The analysis of linear partial differential operators, I and II (Springer, 1983
23. B. Kawohl. Rearrangements and convexity of level sets in PDE (Springer, 1985
24. N. S. Landkof. Foundations of modern potential theory, Grundlehren Der Mathematischen Wissenschaften, vol. 180 (Springer, 1973
25. Y. J. Park. Fractional Polya-Szegö inequality. J. Chungcheong Math. Soc. 24 (2011), 267-271
26. X. Ros-Oton and J. Serra. Fractional Laplacian: Pohozaev identity and nonexistence results. C. R. Math. 350 (2012), 505-508
27. X. Ros-Oton and J. Serra. The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pures Appl. (In the press
28. X. Ros-Oton and J. Serra. The Pohozaev identity for the fractional Laplacian. Preprint, arXiv:1207.5986
29. R. Servadei. The Yamabe equation in a non-local setting. Adv. Nonlin. Analysis 3 (2013), 235-270
30. R. Servadei, A critical fractional Laplace equation in the resonant case. Topolog. Meth. Nonlin. Analysis 43 (2014), 251-267
31. R. Servadei and E. Valdinoci. Mountain pass solutions for non-local elliptic operators. J. Math. Analysis Applic. 389 (2012), 887-898
32. R. Servadei and E. Valdinoci. Variational methods for non-local operators of elliptic type. Discrete Contin. Dynam. Syst. 33 (2013), 2105-2137
33. R. Servadei and E. Valdinoci. A Brezis-Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Analysis 12 (2013), 2445-2464
34. R. Servadei and E. Valdinoci. Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58 (2014), 133-154
35. R. Servadei and E. Valdinoci. The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. (In the press
36. R. Servadei and E. Valdinoci. Fractional Laplacian equations with critical Sobolev exponent. Preprint, available at www.ma.utexas.edu/mp arc-bin/mpa?yn=12-58
37. L. Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. PhD thesis, University of Texas at Austin (2005). (Available at http://www.math. uchicago.edu/'luis/preprints/luisdissreadable.pdf
38. P. R. Stinga and J. L. Torrea. Extension problem and Harnack's inequality for some fractional operators. Commun. PDEs 35 (2010), 2092-2122
39. J. Tan. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. PDEs 36 (2011), 21-41.