Singular solution to Special Lagrangian Equations
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1179-1188.

Nous démontrons l'existence de solutions singulières d'équations speciales lagrangiennes en dimension trois, dans le cas non convexe.

We prove the existence of non-smooth solutions to three-dimensional Special Lagrangian Equations in the non-convex case.

DOI : 10.1016/j.anihpc.2010.05.001
Classification : 35J60, 53C38 
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Nadirashvili, Nikolai; Vlăduţ, Serge. Singular solution to Special Lagrangian Equations. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1179-1188. doi : 10.1016/j.anihpc.2010.05.001. http://www.numdam.org/articles/10.1016/j.anihpc.2010.05.001/

[1] A.D. Alexandrov, Some theorems on partial differential equations of the second order, Vestnik Leningrad. Univ. 9 no. 8 (1954), 3-17

[2] J. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1977), 337-403 | Zbl

[3] J. Bao, J. Chen, B. Guan, M. Ji, Liouville property and regularity of a Hessian quotient equation, Amer. J. Math. 125 no. 2 (2003), 301-316 | Zbl

[4] L. Caffarelli, X. Cabre, Fully Nonlinear Elliptic Equations, Amer. Math. Soc., Providence, RI (1995) | Zbl

[5] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III. Functions of the eigenvalues of the Hessian, Acta Math. 155 no. 3–4 (1985), 261-301 | Zbl

[6] J. Chen, M. Warren, Y. Yuan, A priori estimate for convex solutions to special Lagrangian equations and its application, Comm. Pure Appl. Math. 62 no. 4 (2009), 583-595 | Zbl

[7] R. Courant, D. Hilbert, Methods of Mathematical Physics. Vol. 2, Partial Differential Equations, Wiley (1989) | Zbl

[8] M.G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 no. 1 (1992), 1-67 | Zbl

[9] D. Eisenbud, H. Levin, An algebraic formula for the degree of a C ∞ map germ, Ann. Math. 106 (1977), 19-44 | Zbl

[10] R. Harvey, H.B. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47-157 | Zbl

[11] F.R. Harvey, H.B. Lawson, Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 no. 3 (2009), 396-443 | Zbl

[12] N.V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc. 347 (1995), 857-895 | Zbl

[13] N. Nadirashvili, S. Vlăduţ, Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17 (2007), 1283-1296 | Zbl

[14] N. Nadirashvili, S. Vlăduţ, Singular solutions to fully nonlinear elliptic equations, J. Math. Pures Appl. 89 (2008), 107-113 | Zbl

[15] N. Nadirashvili, S. Vlăduţ, On Hessian fully nonlinear elliptic equations, arXiv:0805.2694 [math.AP] | Zbl

[16] N.S. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 108 no. 1–2 (1988), 57-65 | Zbl

[17] N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Progr. Nonlinear Differential Equations Appl. vol. 2, Birkhäuser Boston, Boston, MA (1989), 939-957

[18] N.S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal. 111 (1990), 153-170

[19] Y. Yuan, A priori estimates for solutions of fully nonlinear special Lagrangian equations, Ann. Inst. H. Pioncaré Anal. Non Linéaire 18 (2001), 261-270 | EuDML | Numdam | Zbl

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