In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equationsIn particular, under the assumption that the Hamiltonian H ∈ C 2 (R n ) is uniformly convex, we prove that D x u and ∂ t u belong to the class SBV loc (Ω).

In this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws We assume that the source term g C1( × × +), that the flux function f C2( × × +) and that ui : fuu(ui,x,t) = 0 is at most countable for every fixed (x,t) Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for (x +h,t), (x,t) in any compact subset of Ω.

Abstract. We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain as a corollary an existence theorem for a related variational problem.

The quantification of post-disturbance root reinforcement (RR) recovery dynamics is of paramount importance for the optimisation of forest ecosystem services and natural hazards risk management in mountain regions. In this work we analyse the long-term root reinforcement dynamic of spruce forests combining data of the Swiss National Forest Inventory with data on root distribution and root mechanical properties. The results show that root reinforcement recovery depends primarily on stand altitude and slope inclination. The maximum root reinforcement recovery rate is reached at circa 100 years. RR increases continuously with different rates for stand ages over 200 years. These results shows that RR in spruce stands varies considerably depending on the local conditions and that its recovery after disturbances requires decades. The new method applied in this study allowed for the first time to quantify the long term dynamics of RR in spruce stands supporting new quantitative approaches for the analysis of shallow landslides disposition in different disturbance regimes of forests.

The authors wish to make the following corrections to this paper [...]