We study Kuzmin's conjecture on the index of nilpotency for the variety N il 5 of associative nil-algebras of degree 5. Due to Vaughan-Lee  the problem is reduced to that for k-generator N il 5-superalgebras, where k ≤ 5. We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.
We prove the unique solvability of a mixed boundary value problem for the p-Laplace operator by means of variational methods. Using the obtained results, we construct an iterative procedure for solving the Cauchy problem for the p-Laplace operator.