We obtain an increase in the exponent of integrability of the derivatives of solutions of two classes of boundary-value problems. We derive estimates of the corresponding norms of the solutions. For a class of quasilinear elliptic systems we establish an L_p-estimate of the gradient of the solutions of class W_m¹, m > 1, p > m, of a boundary-value problem with nonzero condition on the conormal derivative. To solve Signorini's problem we obtain an L_p-estimate, p > 2, of the second derivatives of an L₂-solution with a nonzero one-sided restriction on the conormal derivative. The proof of both results is based on the application of an reverse Hölder inequality with a surface integral established earlier by the author. Bibliography: 5 titles.