Tangents to subsolutions: existence and uniqueness, Part I
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 4, pp. 777-848.

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation f(D 2 u)=0. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the Riesz characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means tangents are always unique and are Riesz kernels) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge–Ampère equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is c>0 is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Hölder continuity of subsolutions when the Riesz characteristic p satisfies 1p<2. Many explicit examples are examined.

The second part of this paper [23] is devoted to the “geometric cases”. A Homogeneity Theorem and an additional Strong Uniqueness Theorem are proved, and the tangents in the Monge–Ampère cases are completely classified.

Il existe une théorie du potentiel intéressante associée à chaque équation, nonlinéaire et élliptique dégénérée, de la forme f(D 2 u)=0. Ceci inclut toutes les théories du potentiel associées aux calibrations. Cet article commence l’étude des tangents aux sous-solutions dans ces théories, un sujet inspiré par l’oeuvre de Kiselman dans le cas pluri-potentiel classique. Fondamentale à notre étude est une nouvelle invariante, la caractéristique de Riesz, qui gouverne les structures asymptotiques. L’existence de tangents aux sous-solutions est établie en général ; on démontre aussi l’existence générale d’une fonction de densité, semi-continue supérieurement. Deux théorèmes qui établissent l’unicité forte de tangents (i.e., tangents sont toujours unique et sont noyaux de Riesz) sont démontrés. Ils comprennent toutes les sous-équations qui sont des cones convexes et O(n)-invariants, ainsi que leurs analogues complexes et quatérnioniques, avec l’exception de l’équation de Monge–Ampère, pour laquelle l’unicité forte ne tient pas. Ils s’appliquent aussi à une grande classe de sous-équations définies géométriquement. Parmi elles sont toutes celles qui proviennent de calibrations. Un résultat de finitude locale, pour les ensembles de densité c>0, est établi dans chaque cas où régit l’unicité forte. Selon un autre résultat, quand la caractéristique de Riesz p satisfait 1p<2, alors toutes les functions sous-harmoniques sont Hölder-continues. On considère beaucoup d’exemples explicites.

La deuxième partie de cet article [23] concerne les “cas géométriques”. On y établit un Théorème d’Homogénéité et un Théorème d’Unicité Forte. Aussi, les espaces tangents pour les équations de Monge–Ampère (réelles, complexes et quaternioniques) sont classifiés complètement.

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DOI: 10.5802/afst.1583

F. Reese Harvey 1; H. Blaine Lawson 2

1 Department of Mathematics, RICE University, Houston, TX 77005-1982, USA
2 Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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F. Reese Harvey; H. Blaine Lawson. Tangents to subsolutions: existence and uniqueness, Part I. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 4, pp. 777-848. doi : 10.5802/afst.1583. https://afst.centre-mersenne.org/articles/10.5802/afst.1583/

[1] Scott N. Armstrong; Charles K. Smart; Boyan Sirakov Fundamental solutions of homogeneous fully nonlinear elliptic equations, Commun. Pure Appl. Math., Volume 64 (2011) no. 6, pp. 737-777 | MR | Zbl

[2] Eric Bedford; Bert Alan Taylor The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 | Zbl

[3] Enrico Bombieri Algebraic values of meromorphic maps, Invent. Math., Volume 10 (1970), pp. 267-287 addendum in ibid. 11 (1970), p. 163-166 | DOI | MR

[4] Luis A. Caffarelli; Xavier Cabré Fully Nonlinear Elliptic Equations, Colloquium Publications, 43, American Mathematical Society, 1995, v+104 pages | MR | Zbl

[5] Michael G. Crandall Viscosity solutions: a primer, Viscosity solutions and applications (Lecture Notes in Mathematics), Volume 1660, Springer, 1997, pp. 1-43 | DOI | MR | Zbl

[6] Michael G. Crandall; Hitoshi Ishii; Pierre-Louis Lions User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67 | DOI | MR | Zbl

[7] Jean-Pierre Demailly Complex analytic and differential geometry (http://www-fourier.ujf-grenoble.fr/~demailly/documents.html)

[8] Lawrence C. Evans Regularity for fully nonlinear elliptic equations and motion by mean curvature, Viscosity solutions and applications (Lecture Notes in Mathematics), Volume 1660, Springer, 1997, pp. 97-133 | MR | Zbl

[9] Herbert Federer Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, 1969, xiv+676 pages | MR | Zbl

[10] Lars Gårding An inequality for hyperbolic polynomials, J. Math. Mech., Volume 8 (1959) no. 2, pp. 957-965 | MR | Zbl

[11] F. Reese Harvey Removable singularities and structure theorems for positive currents, Partial differential equations (Proc. Sympos. Pure Math.), Volume 23 (1973), pp. 123-133 | Zbl

[12] F. Reese Harvey; H. Blaine Lawson Subequation characterizations of various classical functions (in preparation)

[13] F. Reese Harvey; H. Blaine Lawson Calibrated geometries, Acta Math., Volume 148 (1982), pp. 47-157 | DOI | MR | Zbl

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

[15] F. Reese Harvey; H. Blaine Lawson Hyperbolic polynomials and the Dirichlet problem (2009) (https://arxiv.org/abs/0912.5220)

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

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

[18] F. Reese Harvey; H. Blaine Lawson Plurisubharmonicity in a general geometric context, Geometry and analysis, No. 1 (Advanced Lectures in Mathematics), Volume 17, International Press; Higher Education Press, 2011, pp. 363-401 | MR | Zbl

[19] F. Reese Harvey; H. Blaine Lawson The AE Theorem and addition theorems for quasi-convex functions (2013) (https://arxiv.org/abs/1309.1770)

[20] F. Reese Harvey; H. Blaine Lawson The equivalence of viscosity and distributional subsolutions for convex subequations – the strong Bellman principle, Bull. Braz. Math. Soc., Volume 44 (2013) no. 4, pp. 621-652 | DOI | MR | Zbl

[21] F. Reese Harvey; H. Blaine Lawson Gårding’s theory of hyperbolic polynomials, Commun. Pure Appl. Math., Volume 66 (2013) no. 7, pp. 1102-1128 | DOI | Zbl

[22] F. Reese Harvey; H. Blaine Lawson p-convexity, p-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., Volume 62 (2013) no. 1, pp. 149-169 | DOI | MR | Zbl

[23] F. Reese Harvey; H. Blaine Lawson Tangents to subsolutions – existence and uniqueness, Part II (2014) (https://arxiv.org/abs/1408.5851) | Zbl

[24] F. Reese Harvey; H. Blaine Lawson The Dirichlet Problem with Prescribed Interior Singularities, Adv. Math., Volume 303 (2016), pp. 1319-1357 | MR | Zbl

[25] Lars Hörmander An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library, 7, North-Holland, 1990, xii+254 pages | MR | Zbl

[26] Lars Hörmander Notions of Convexity, Progress in Mathematics, 127, Birkhäuser, 1994, viii+414 pages | MR | Zbl

[27] Lars Hörmander; Ragnar Sigurdsson Limit sets of plurisubharmonic functions, Math. Scand., Volume 65 (1989) no. 2, pp. 308-320 | DOI | MR | Zbl

[28] Hitoshi Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s, Commun. Pure Appl. Math., Volume 42 (1989) no. 1, pp. 15-45 | DOI | MR | Zbl

[29] Christer O. Kiselman Tangents of plurisubharmonic functions, International Symposium in Memory of Hua Loo Keng, Vol. II (Beijing, 1988), Springer, 1991, pp. 157-167 | Zbl

[30] Christer O. Kiselman Plurisubharmonic functions and potential theory in several complex variables, Development of mathematics 1950-2000, Birkhäuser, 2000, pp. 655-714 | DOI | MR | Zbl

[31] Maciej Klimek Pluripotential theory, London Mathematical Society Monographs, New Series, 6, Clarendon Press, 1991, xiv+266 pages | MR | Zbl

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

[33] Denis A. Labutin Isolated singularities for fully nonlinear elliptic equations, J. Differ. Equations, Volume 177 (2001) no. 1, pp. 49-76 | DOI | MR | Zbl

[34] Denis A. Labutin Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., Volume 111 (2002) no. 1, pp. 1-49 | DOI | MR | Zbl

[35] Denis A. Labutin Singularities of viscosity solutions of fully nonlinear elliptic equations, Viscosity Solutions of Differential Equations and Related Topics (RIMS Kôkyûroku), Volume 1287, Kyoto University, 2002, pp. 45-57 | MR

[36] N. S. Landkof Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, 180, Springer, 1972, x+424 pages | MR | Zbl

[37] Jiping Sha p-convex riemannian manifolds, Invent. Math., Volume 83 (1986), pp. 437-447 | MR | Zbl

[38] Ragnar Sigurdsson Growth properties of analytic and plurisubharmonic functions of finite order, Math. Scand., Volume 59 (1986) no. 2, pp. 235-304 | DOI | MR | Zbl

[39] Leon Simon Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Australian National University, 1983, vii+272 pages | MR | Zbl

[40] Yum-Tong Siu Analyticity of sets associated to Lelong numbers and the extension on closed positive currents, Invent. Math., Volume 27 (1974), pp. 53-156 | MR | Zbl

[41] Neil S. Trudinger Hölder gradient estimates for fully nonlinear equations, Proc. R. Soc. Edinb., Sect. A, Volume 108 (1988) no. 1-2, pp. 57-65 | DOI | Zbl

[42] Neil S. Trudinger; Xu-Jia Wang Hessian measures. I, Topol. Methods Nonlinear Anal., Volume 10 (1997) no. 2, pp. 225-239 | DOI | MR | Zbl

[43] Neil S. Trudinger; Xu-Jia Wang Hessian measures. II, Ann. Math., Volume 150 (1999) no. 2, pp. 579-604 | DOI | MR | Zbl

[44] Neil S. Trudinger; Xu-Jia Wang Hessian measures. III, J. Funct. Anal., Volume 193 (2002) no. 1, pp. 1-23 | DOI | MR | Zbl

[45] Hung-Hsi Wu Manifolds of partially positive curvature, Indiana Univ. Math. J., Volume 36 (1987) no. 3, pp. 523-548 | MR

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