The Restriction Theorem for Fully Nonlinear Subequations

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X ⊂ Z determine a restricted subequation on X . The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy the same constraints. We first prove an elementary 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.