Elliptic problems with integral diffusion
Yannick Sire1
1 Laboratoire LATP CNRS UMR 7353 Université Aix-Marseille 13397 Marseille Cedex 20 France
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 21, 10 p.

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.

DOI : 10.5802/slsedp.16
Yannick Sire 1

1 Laboratoire LATP CNRS UMR 7353 Université Aix-Marseille 13397 Marseille Cedex 20 France 
@article{SLSEDP_2011-2012____A21_0,
     author = {Yannick Sire},
     title = {Elliptic problems with integral diffusion},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:21},
     pages = {1--10},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2011-2012},
     doi = {10.5802/slsedp.16},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.16/}
}
TY  - JOUR
AU  - Yannick Sire
TI  - Elliptic problems with integral diffusion
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:21
PY  - 2011-2012
SP  - 1
EP  - 10
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.16/
DO  - 10.5802/slsedp.16
LA  - en
ID  - SLSEDP_2011-2012____A21_0
ER  - 
%0 Journal Article
%A Yannick Sire
%T Elliptic problems with integral diffusion
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:21
%D 2011-2012
%P 1-10
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.16/
%R 10.5802/slsedp.16
%G en
%F SLSEDP_2011-2012____A21_0
Yannick Sire. Elliptic problems with integral diffusion. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 21, 10 p. doi : 10.5802/slsedp.16. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.16/

[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 | Zbl

[AC00] Luigi Ambrosio and Xavier Cabré. Entire solutions of semilinear elliptic equations in ℝ 3 and a conjecture of De Giorgi. J. Amer. Math. Soc., 13(4):725–739, 2000. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[CRS10] L. Caffarelli, J.-M. Roquejoffre, and O. Savin. Nonlocal minimal surfaces. Comm. Pure Appl. Math., 63(9):1111–1144, 2010. | MR | Zbl

[CS07] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Commun. in PDE, 32(8):1245, 2007. | MR | Zbl

[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≥9. To appear Ann. of Maths, 2011. | Zbl

[dMMS10] González Maria del Mar, Rafe Mazzeo, and Yannick Sire. Singular solutions of fractional order conformal laplacians. To appear J. Geom. Anal., 2010. | MR

[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 | Zbl

[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 | Zbl

[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 | Zbl

[Gon05] María del Mar González. Singular sets of a class of locally conformally flat manifolds. Duke Math. J., 129(3):551–572, 2005. | MR | Zbl

[GZ03] C. Robin Graham and Maciej Zworski. Scattering matrix in conformal geometry. Invent. Math., 152(1):89–118, 2003. | MR | Zbl

[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 | Zbl

[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 | Zbl

[Lab03] Denis A. Labutin. Wiener regularity for large solutions of nonlinear equations. Ark. Mat., 41(2):307–339, 2003. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[SY88] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math., 92(1):47–71, 1988. | MR | Zbl

Cité par Sources :