Advertisement

Sharp energy estimates for nonlinear fractional diffusion equations

  • Xavier CabréEmail author
  • Eleonora Cinti
Article

Abstract

We study the nonlinear fractional equation \((-\Delta )^su=f(u)\) in \(\mathbb R ^n,\) for all fractions \(0<s<1\) and all nonlinearities \(f\). For every fractional power \(s\in (0,1)\), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension \(n=3\) whenever \(1/2\le s<1\). This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation \(-\Delta u=f(u)\) in \(\mathbb R ^n\). It remains open for \(n=3\) and \(s<1/2\), and also for \(n\ge 4\) and all \(s\).

Keywords

Fractional laplacian Energy estimates Symmetry properties 

Mathematics Subject Classification

35R11 35B06 35B08 

Notes

Acknowledgments

Both authors were supported by grants MINECO MTM2011-27739-C04-01 (Spain) and GENCAT 2009SGR345 (Catalunya). The second author was partially supported by University of Bologna (Italy), funds for selected research topics, and by the ERC Starting Grant “AnOptSetCon” n. 258685.

References

  1. 1.
    Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65, 9–33 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alberti, G., Bouchitté, G., Seppecher, S.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144, 1–46 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in \(re^3\) and a conjecture of De Giorgi. J. Am. Math. Soc. 13, 725–739 (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cabré, X., Cinti, E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst. 28, 1179–1206 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Preprint, arXiv: 1012.0867Google Scholar
  6. 6.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. To appear in Trans. Am. Math. Soc. Preprint, arXiv: 1111.0796Google Scholar
  7. 7.
    Cabré, X., Solà-Morales, J.: Layer solutions in a halph-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Diff. Eq. 32, 1245–1260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Caffarelli, L., Souganidis, P.E.: Convergence of nonlocal threshold dynamics approximations to front propagation. Arch. Rational Mech. Anal. 195, 1–23 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Caffarelli, L., Valdinoci, E.: Regularity properties of nonlocal minimal surfaces via limiting arguments. preprint, arXiv: 1105.1158Google Scholar
  12. 12.
    Cinti, E.: Bistable Elliptic Equations With Fractional Diffusion. Ph.D. Thesis, Universitat Politècnica de Catalunya and Università di Bologna (2010)Google Scholar
  13. 13.
    del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi Conjecture in dimension \(N\ge 9\). Annal. Math. 174, 1485–1569 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7, 77–116 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Frank R. L., Lenzmann, E.: Uniqueness and nondegeneracy of ground states for \((-\Delta )^sQ+Q-Q^{\alpha +1}=0\) in \(re\). Preprint, arXiv: 1009.4042Google Scholar
  16. 16.
    Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    González, MdM: Gamma convergence of an energy functional related to the fractional Laplacian. Calc. Var. 36, 173–210 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, New York (1972)Google Scholar
  19. 19.
    Moschini, L.: New Liouville theorems for linear second order degenerate elliptic equations in divergence form. Ann. I. H. Poincaré 22, 11–23 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nekvinda, A.: Characterization of traces of the weighted Sobolev space \(H^{1, p}_{\varepsilon, M}\). Funct. Approx. Comment. Math. 20, 143–151 (1992)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Nekvinda, A.: Characterization of traces of the weighted Sobolev space \(W^{1, p}(\Omega, d_M^{\varepsilon })\) on \(M\). Czechoslovak Math. J. 43, 713–722 (1993)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Annali di Matematica Pura ed Applicata (2012). doi: 10.1007/s10231-011-0243-9
  23. 23.
    Savin, O.: Phase ransitions: regularity of flat level sets. Ann. Math. 169, 41–78 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. Preprint, http://arxiv.org/abs/1007.2114
  25. 25.
    Savin, O., Valdinoci, E.: \(\Gamma \)-convergence for nonlocal phase transitions. Preprint, http://arxiv.org/abs/1007.1725
  26. 26.
    Savin, O., Valdinoci, E.: Regularity of nonlocal minimal cones in dimension 2. Preprint, http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-8
  27. 27.
    Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)Google Scholar
  28. 28.
    Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256, 1842–1864 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35, 2092–2122 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ICREA and Universitat Politècnica de CatalunyaDepartament de Matemàtica Aplicada 1BarcelonaSpain
  2. 2.Dipartimento di MatematicaUniversità degli Studi di Bologna BolognaItaly

Personalised recommendations