Variational Problem with Degeneracy at the Boundary of the Interval

The weighted $L_p$-norms of derivatives are estimated in terms of the weighted $L_p$-norm of the highest derivative and the traces of the function and its derivatives at the given points of closure of the bounded interval; weights are powers of the distance to the nearest endpoint of the interval. For functions with zero traces, sharper estimates are established. For the integral quadratic functional with degenerate coefficients, we prove the existence and uniqueness of the solution to the problem of minimization of a functional on a function class with zero traces.