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 \(s\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Alberti, G., Bouchitté, G., Seppecher, S.: Phase transition with the line-tension effect. Arch. Rational Mech. Anal. 144, 1–46 (1998)
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)
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)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Preprint, arXiv: 1012.0867
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: 1111.0796
Cabré, X., Solà-Morales, J.: Layer solutions in a halph-space for boundary reactions. Commun. Pure Appl. Math. 58, 1678–1732 (2005)
Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Diff. Eq. 32, 1245–1260 (2007)
Caffarelli, L., Souganidis, P.E.: Convergence of nonlocal threshold dynamics approximations to front propagation. Arch. Rational Mech. Anal. 195, 1–23 (2010)
Caffarelli, L., Valdinoci, E.: Regularity properties of nonlocal minimal surfaces via limiting arguments. preprint, arXiv: 1105.1158
Cinti, E.: Bistable Elliptic Equations With Fractional Diffusion. Ph.D. Thesis, Universitat Politècnica de Catalunya and Università di Bologna (2010)
del Pino, M., Kowalczyk, M., Wei, J.: On De Giorgi Conjecture in dimension \(N\ge 9\). Annal. Math. 174, 1485–1569 (2011)
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)
Frank R. L., Lenzmann, E.: Uniqueness and nondegeneracy of ground states for \((-\Delta )^sQ+Q-Q^{\alpha +1}=0\) in \(re\). Preprint, arXiv: 1009.4042
Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problems. Math. Ann. 311, 481–491 (1998)
González, MdM: Gamma convergence of an energy functional related to the fractional Laplacian. Calc. Var. 36, 173–210 (2009)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, New York (1972)
Moschini, L.: New Liouville theorems for linear second order degenerate elliptic equations in divergence form. Ann. I. H. Poincaré 22, 11–23 (2005)
Nekvinda, A.: Characterization of traces of the weighted Sobolev space \(H^{1, p}_{\varepsilon, M}\). Funct. Approx. Comment. Math. 20, 143–151 (1992)
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)
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
Savin, O.: Phase ransitions: regularity of flat level sets. Ann. Math. 169, 41–78 (2009)
Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. Preprint, http://arxiv.org/abs/1007.2114
Savin, O., Valdinoci, E.: \(\Gamma \)-convergence for nonlocal phase transitions. Preprint, http://arxiv.org/abs/1007.1725
Savin, O., Valdinoci, E.: Regularity of nonlocal minimal cones in dimension 2. Preprint, http://www.ma.utexas.edu/mparc-bin/mpa?yn=12-8
Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)
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)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35, 2092–2122 (2010)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by O.Savin.
Rights and permissions
About this article
Cite this article
Cabré, X., Cinti, E. Sharp energy estimates for nonlinear fractional diffusion equations. Calc. Var. 49, 233–269 (2014). https://doi.org/10.1007/s00526-012-0580-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0580-6