The restriction theorem for fully nonlinear subequations  [ Le théorème de restriction pour sous-équations complètement non-linéaires ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 1, p. 217-265
Soit X une sous-variété d’une variété Z. On se pose la question  : sous quelles conditions est-il vrai que les sous-solutions de viscosité d’une équation aux derivées partielles complètement non-linéaires sur Z, restreintes à X, sont des sous-solutions de viscosité de l’équation induite sur X  ? D’abord on démontre un résultat de base qui s’applique aux équations générales. Ensuite, deux résultats définitifs sont établis. Le premier s’applique à toutes les équations qui sont “définies géométriquement” et le deuxième s’applique aux équations qui peuvent être transformées par jet-équivalence en modèle de coefficients constants (i.e., modèle euclidien). En conséquence, nous obtenons une longue liste de cas intéressants du point du vue géométrique et analytique, où la réponse à notre question est positive.
Let X be a submanifold of a manifold Z. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on Z, restrict to be viscosity subsolutions of the restricted subequation on X? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed to a constant coefficient (i.e., euclidean) model. This provides a long list of geometrically and analytically interesting cases where restriction holds.
DOI : https://doi.org/10.5802/aif.2846
Classification:  35J25,  35J70,  32W20,  32U05,  53C38
Mots clés: solution de viscosité, sous-solution de viscosité, équations elliptiques non-linéaires de second ordre, restriction, sous-variété, théorie pluripotentielle
@article{AIF_2014__64_1_217_0,
     author = {Harvey, F. Reese and Lawson, H. Blaine, Jr.},
     title = {The restriction theorem for fully nonlinear subequations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     pages = {217-265},
     doi = {10.5802/aif.2846},
     zbl = {1320.32037},
     mrnumber = {3330548},
     zbl = {06387273},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_1_217_0}
}
Harvey, F. Reese; Lawson, H. Blaine Jr. The restriction theorem for fully nonlinear subequations. Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 217-265. doi : 10.5802/aif.2846. http://www.numdam.org/item/AIF_2014__64_1_217_0/

[1] Alesker, Semyon Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, Bull. Sci. Math., Tome 127 (2003) no. 1, pp. 1-35 | Article | MR 1957796 | Zbl 1033.15013

[2] Alesker, Semyon Quaternionic Monge-Ampère equations, J. Geom. Anal., Tome 13 (2003) no. 2, pp. 205-238 | Article | MR 1967025 | Zbl 1058.32028

[3] Alesker, Semyon; Verbitsky, Misha Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, J. Geom. Anal., Tome 16 (2006) no. 3, pp. 375-399 | Article | MR 2250051 | Zbl 1106.32023

[4] Alexandrov, A. D. The Dirichlet problem for the equation Detz i,j =ψ(z 1 ,...,z n ,x 1 ,...,x n ), I. Vestnik, Leningrad Univ., Tome 13 (1958) no. 1, pp. 5-24

[5] Bedford, Eric; Taylor, B. A. The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Tome 37 (1976) no. 1, pp. 1-44 | Article | MR 445006 | Zbl 0315.31007

[6] Bremermann, H. J. On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries, Trans. Amer. Math. Soc., Tome 91 (1959), pp. 246-276 | MR 136766 | Zbl 0091.07501

[7] Crandall, Michael G. Viscosity solutions: a primer, Viscosity solutions and applications (Montecatini Terme, 1995), Springer, Berlin (Lecture Notes in Math.) Tome 1660 (1997), pp. 1-43 | MR 1462699 | Zbl 0901.49026

[8] Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Tome 27 (1992) no. 1, pp. 1-67 | Article | MR 1118699 | Zbl 0755.35015

[9] Harvey, F. Reese; Lawson, H. Blaine Jr. Hyperbolic polynomials and the Dirichlet problem (ArXiv:0912.5220)

[10] Harvey, F. Reese; Lawson, H. Blaine Jr. Potential theory on almost complex manifolds (Ann. Inst. Fourier (to appear). ArXiv:1107.2584)

[11] Harvey, F. Reese; Lawson, H. Blaine Jr. Calibrated geometries, Acta Math., Tome 148 (1982), pp. 47-157 | Article | MR 666108 | Zbl 0584.53021

[12] Harvey, F. Reese; Lawson, H. Blaine Jr. Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., Tome 62 (2009) no. 3, pp. 396-443 | Article | MR 2487853 | Zbl 1173.35062

[13] Harvey, F. Reese; Lawson, H. Blaine Jr. Duality of positive currents and plurisubharmonic functions in calibrated geometry, Amer. J. Math., Tome 131 (2009) no. 5, pp. 1211-1239 | Article | MR 2555839 | Zbl 1179.53058

[14] Harvey, F. Reese; Lawson, H. Blaine Jr. An introduction to potential theory in calibrated geometry, Amer. J. Math., Tome 131 (2009) no. 4, pp. 893-944 | Article | MR 2543918 | Zbl 1170.53031

[15] Harvey, F. Reese; Lawson, H. Blaine Jr. Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds, J. Differential Geom., Tome 88 (2011) no. 3, pp. 395-482 | MR 2844439 | Zbl 1235.53042

[16] Harvey, F. Reese; Lawson, H. Blaine Jr. Plurisubharmonicity in a general geometric context, Geometry and analysis. No. 1, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 17 (2011), pp. 363-402 | Zbl 1271.31011

[17] Harvey, F. Reese; Lawson, H. Blaine Jr.; Cao, H.-D.; Eds., S.-T. Yau Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in Geometry, International Press, Sommerville, MA, Tome 18 (2013), pp. 103-156 | Zbl pre06296845

[18] Hunt, L. R.; Murray, John J. q-plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J., Tome 25 (1978) no. 3, pp. 299-316 | Article | MR 512901 | Zbl 0378.32013

[19] Krylov, N. V. On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc., Tome 347 (1995) no. 3, pp. 857-895 | Article | MR 1284912 | Zbl 0832.35042

[20] Lawson, H. Blaine Jr. Lectures on minimal submanifolds. Vol. I, Publish or Perish Inc., Wilmington, Del., Mathematics Lecture Series, Tome 9 (1980), pp. iv+178 | MR 576752 | Zbl 0434.53006

[21] Nijenhuis, Albert; Woolf, William B. Some integration problems in almost-complex and complex manifolds., Ann. of Math. (2), Tome 77 (1963), pp. 424-489 | Article | MR 149505 | Zbl 0115.16103

[22] Pali, Nefton Fonctions plurisousharmoniques et courants positifs de type (1,1) sur une variété presque complexe, Manuscripta Math., Tome 118 (2005) no. 3, pp. 311-337 | Article | MR 2183042 | Zbl 1089.32033

[23] Slodkowski, Zbigniew The Bremermann-Dirichlet problem for q-plurisubharmonic functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 11 (1984) no. 2, pp. 303-326 | Numdam | MR 764948 | Zbl 0583.32046