On construction of weak solutions with higher integrable gradients to linear hyperbolic partial differential systems

Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic differential equations. Bibliography: 6 titles.