Center for diffusion of mathematic journals


Séminaire Laurent Schwartz — EDP et applications

Table of contents for this volume | Previous article | Next article
Yannick Sire
Elliptic problems with integral diffusion
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exp. No. 21, 10 p., doi: 10.5802/slsedp.16
Article PDF

Résumé - Abstract

In this paper, we review several recent results dealing with elliptic equations with non local diffusion. More precisely, we investigate several problems involving the fractional laplacian. Finally, we present a conformally covariant operator and the associated singular and regular Yamabe problem.


[AAC01] Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math., 65(1-3):9–33, 2001. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday.  MR 1843784 |  Zbl 1121.35312
[AC00] Luigi Ambrosio and Xavier Cabré. Entire solutions of semilinear elliptic equations in $ \mathbb{R}^3$ and a conjecture of De Giorgi. J. Amer. Math. Soc., 13(4):725–739, 2000.  MR 1775735 |  Zbl 0968.35041
[BM09] Isabeau Birindelli and Rafe Mazzeo. Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space. Indiana Univ. Math. J., 58(5):2347–2368, 2009.  MR 2583503 |  Zbl 1183.35136
[CC10] Xavier Cabré and Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete Contin. Dyn. Syst., 28(3):1179–1206, 2010.  MR 2644786 |  Zbl 1193.35242
[CDDS11] Antonio Capella, Juan Dávila, Louis Dupaigne, and Yannick Sire. Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differential Equations, 36(8):1353–1384, 2011.  MR 2825595 |  Zbl 1231.35076
[CHY04] Sun-Yung A. Chang, Fengbo Hang, and Paul C. Yang. On a class of locally conformally flat manifolds. Int. Math. Res. Not., (4):185–209, 2004.  MR 2040327 |  Zbl 1137.53327
[CRS10] L. Caffarelli, J.-M. Roquejoffre, and O. Savin. Nonlocal minimal surfaces. Comm. Pure Appl. Math., 63(9):1111–1144, 2010.  MR 2675483 |  Zbl 1248.53009
[CS07] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Commun. in PDE, 32(8):1245, 2007.  MR 2354493 |  Zbl 1143.26002
[CS10] X. Cabré and Y. Sire. Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Preprint, 2010.
[dKW11] Manuel del, Mike Kowalczyk, and Juncheng Wei. On De Giorgi Conjecture in Dimension ${N} \ge 9$. To appear Ann. of Maths, 2011.  Zbl 1238.35019
[dMMS10] González Maria del Mar, Rafe Mazzeo, and Yannick Sire. Singular solutions of fractional order conformal laplacians. To appear J. Geom. Anal., 2010.  MR 2927681 |  Zbl pre06112733
[DS10] Louis Dupaigne and Yannick Sire. A Liouville theorem for non local elliptic equations. In Symmetry for elliptic PDEs, volume 528 of Contemp. Math., pages 105–114. Amer. Math. Soc., Providence, RI, 2010.  MR 2759038 |  Zbl 1218.35243
[FSV11] Alberto Farina, Yannick Sire, and Enrico Valdinoci. Stable solutions on Riemannian manifolds. To appear J. of Geom. Anal., 2011.
[GG98] N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems. Math. Ann., 311(3):481–491, 1998.  MR 1637919 |  Zbl 0918.35046
[GJMS92] C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling. Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2), 46(3):557–565, 1992.  MR 1190438 |  Zbl 0726.53010
[Gon05] María del Mar González. Singular sets of a class of locally conformally flat manifolds. Duke Math. J., 129(3):551–572, 2005.  MR 2169873 |  Zbl 1088.53023
[GZ03] C. Robin Graham and Maciej Zworski. Scattering matrix in conformal geometry. Invent. Math., 152(1):89–118, 2003.  MR 1965361 |  Zbl 1030.58022
[JL73] D. D. Joseph and T. S. Lundgren. Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal., 49:241–269, 1972/73.  MR 340701 |  Zbl 0266.34021
[Juh] A. Juhl. On conformally covariant powers of the laplacian. Preprint.
[Juh09] Andreas Juhl. Families of conformally covariant differential operators, $Q$-curvature and holography, volume 275 of Progress in Mathematics. Birkhäuser Verlag, Basel, 2009.  MR 2521913 |  Zbl 1177.53001
[Lab03] Denis A. Labutin. Wiener regularity for large solutions of nonlinear equations. Ark. Mat., 41(2):307–339, 2003.  MR 2011924 |  Zbl 1071.35048
[MP96] Rafe Mazzeo and Frank Pacard. A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. J. Differential Geom., 44(2):331–370, 1996.  MR 1425579 |  Zbl 0869.35040
[QG10] J. Qing and M.d.M. Gonzalez. the fractional yamabe problem. Preprint, 2010.
[Sav09] Ovidiu Savin. Regularity of flat level sets in phase transitions. Ann. of Math. (2), 169(1):41–78, 2009.  MR 2480601 |  Zbl 1180.35499
[SV09a] Yannick Sire and Enrico Valdinoci. Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result. J. Funct. Anal., 256(6):1842–1864, 2009.  MR 2498561 |  Zbl 1163.35019
[SV09b] Yannick Sire and Enrico Valdinoci. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal., 256(6):1842–1864, 2009.  MR 2498561 |  Zbl 1163.35019
[SV10a] Ovidiu Savin and Enrico Valdinoci. A gamma-convergence result for non local phase transitions. Preprint, 2010.
[SV10b] Yannick Sire and Enrico Valdinoci. Some elliptic PDEs on Riemannian manifolds with boundary. Pacific J. Math., 248(2):475–492, 2010.  MR 2741258 |  Zbl 1205.35061
[SY88] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math., 92(1):47–71, 1988.  MR 931204 |  Zbl 0658.53038
Copyright Cellule MathDoc 2019 | Credit | Site Map