Local Flexibility for Open Partial Differential Relations

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended global provided all but the highest derivatives stay constant along subset. The applicability this general result is illustrated by a number examples, dealing with convex embeddings hypersurfaces, forms, and lapse functions in Lorentzian geometry. main application approximation sections which have very restrictive properties an dense subsets. This shows, for instance, given any $K \in \mathbb{R}$ every manifold dimension at least two carries complete $C^{1,1}$-metric which, on subset, smooth sectional curvature $K$. Of course impossible $C^2$-metrics general.