Tangents to subsolutions: existence and uniqueness, Part I

Harvey, F. Reese; Lawson, H. Blaine

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 777-848.

Il existe une théorie du potentiel intéressante associée à chaque équation, nonlinéaire et élliptique dégénérée, de la forme $f\left(D^{2}u\right)=0$. 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$\left(n\right)$-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 $\ge c>0$, est établi dans chaque cas où régit l’unicité forte. Selon un autre résultat, quand la caractéristique de Riesz $p$ satisfait $1\le p<2$, 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.

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f\left(D^{2}u\right)=0$. 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$\left(n\right)$-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 $\ge c>0$ 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 $1\le p<2$. 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.

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