Regularity of the Minimum Time and of Viscosity Solutions of Degenerate Eikonal Equations via Generalized Lie Brackets

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Ho ̈lder regularity of the minimum time function in optimal control. We then apply such result to prove Ho ̈lder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Ho ̈lder regularity are essentially necessary, at least for smooth vector fields and target.