2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102012We analyse the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of independent interest. As an application we prove a conjecture of Ennio De Giorgi on the evolution of submanifolds of the Euclidean space by the gradient of functionals depending on the derivatives of the distance function.Analysis of PDEsFunctional Analysis53A07; 53A55text