Quasilinear elliptic and parabolic equations of arbitrary order

This paper is a survey of recent results on the solution of boundary value problems for quasilinear elliptic and parabolic equations of order $2m$, of divergent form. The main results in this direction were first obtained in 1961 by Vishik, Browder, the author and others, and are presented in the first part of the paper. We also indicate the spaces in which the elliptic and parabolic operators induce homeomorphisms in the strongly elliptic case. When the variation of the operator is merely semibounded below, the Dirichlet problem is soluble for any right-hand side, though not uniquely. In the second part we present the work of several authors concerning the solution of operator equations in Banach spaces, among them Minty, Browder, Leray, Lions, Dubinskii, Pokhozhaev. The results are then applied to non-linear differential equations.