We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.

We study the formation of singularities for hypersurfaces evolving by mean curvature. After recalling the basic properties of the flow and the classical results about curves and convex surfaces, we give account of some recent developments of the theory for the case of surfaces with positive mean curvature. We show that for such surfaces we can obtain a–priori estimates on the principal curvatures which enable us to classify the singular profiles by the use of rescaling techniques.

We consider an optimal control problem of Mayer type and prove that,
under suitable conditions on the system, the value function is
differentiable along optimal trajectories, except possibly at the
endpoints. We provide counterexamples to show that this property may fail
to hold if some of our conditions are violated. We then apply our regularity
result to derive optimality conditions for the trajectories of the system.

We consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature...

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