We prove that a harmonic map with small energy and monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density; moreover, the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability. This work generalizes the results in [CL][DF][LG12], which were proved under the assumptions that the singular sets are...

We study an optimal control problem in Bolza form and we consider the value function associated to this problem. We prove two verification theorems which ensure that, if a function W satisfies some suitable weak continuity assumptions and a Hamilton-Jacobi-Bellman inequality outside a countably H-rectifiable set, then it is lower or equal to the value function. These results can be used for opt...