Introduction

Abstract

The principal objective of this work is the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process. This means we shall be concerned with the solvability of boundary value problems (primarily the Dirichlet problem) and related general properties of solutions of linear equations

$$ Lu \equiv {a^{{ij}}}(x){D_{{ij}}}u + {b^i}(x){D_i}u + c(x)u = f(x),i,j = 1,2,.....,n, $$

((1.1))

and of quasilinear equations

$$ Qu \equiv {a^{{ij}}}(x,u,Du){D_{{ij}}}u + b(x,u,Du) = 0 $$

((1.2))

. Here Du = (D ₁u,..., D _nu), where D _iu = ∂u/∂x _i, D _iju = ∂ ²u/∂x _i ∂x _j, etc., and the summation convention is understood. The ellipticity of these equations is expressed by the fact that the coefficient matrix [a ^ij] is (in each case) positive definite in the domain of the respective arguments. We refer to an equation as uniformly elliptic if the ratio γ of maximum to minimum eigenvalue of the matrix [a ^ij] is bounded. We shall be concerned with both non-uniformly and uniformly elliptic equations.