The restriction theorem for fully nonlinear subequations

Harvey, F. Reese; Lawson, H. Blaine Jr.

Annales de l'Institut Fourier, Tome 64 (2014) no. 1, pp. 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.

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