The restriction theorem for fully nonlinear subequations
Annales de l'Institut Fourier, Volume 64 (2014) no. 1, pp. 217-265.

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.

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.

DOI: 10.5802/aif.2846
Classification: 35J25, 35J70, 32W20, 32U05, 53C38
Keywords: Viscosity solution, viscosity subsolution, nonlinear second-order elliptic equations, restriction, submanifold, pluripotential theory
Mot clés : solution de viscosité, sous-solution de viscosité, équations elliptiques non-linéaires de second ordre, restriction, sous-variété, théorie pluripotentielle
Harvey, F. Reese 1; Lawson, H. Blaine Jr. 2

1 Rice University Department of Mathematics P.O. Box 1892 MS-172 Houston, 77251(Texas)
2 Stony Brook University Department of Mathematics Stony Brook NY, 11794-3651 (USA)
@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},
     pages = {217--265},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {1},
     year = {2014},
     doi = {10.5802/aif.2846},
     zbl = {1320.32037},
     mrnumber = {3330548},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2846/}
}
TY  - JOUR
AU  - Harvey, F. Reese
AU  - Lawson, H. Blaine Jr.
TI  - The restriction theorem for fully nonlinear subequations
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 217
EP  - 265
VL  - 64
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2846/
DO  - 10.5802/aif.2846
LA  - en
ID  - AIF_2014__64_1_217_0
ER  - 
%0 Journal Article
%A Harvey, F. Reese
%A Lawson, H. Blaine Jr.
%T The restriction theorem for fully nonlinear subequations
%J Annales de l'Institut Fourier
%D 2014
%P 217-265
%V 64
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2846/
%R 10.5802/aif.2846
%G en
%F 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, Volume 64 (2014) no. 1, pp. 217-265. doi : 10.5802/aif.2846. https://aif.centre-mersenne.org/articles/10.5802/aif.2846/

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

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

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

[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., Volume 13 (1958) no. 1, pp. 5-24

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

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

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

[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.), Volume 27 (1992) no. 1, pp. 1-67 | DOI | MR | Zbl

[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., Volume 148 (1982), pp. 47-157 | DOI | MR | Zbl

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

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

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

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

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

[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, Volume 18, International Press, Sommerville, MA, 2013, pp. 103-156

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

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

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

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

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

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

Cited by Sources: