The Dirichlet Problem for Second-Order Quasi-Linear Nonuniformly Elliptic Equations

Abstract

We wish to investigate the following equation in the bounded domain Ω < E_n, n ≥ π:

$${A_{ij}}\left( {x,u,{u_x}} \right){u_{ij}} = \beta \left( {x,u,{u_x}} \right),$$

(1)

in which ${A_{ij}} = {A_{ji}},{u_x} = \left( {{u_{{x_1}}}, \ldots {u_{xn}}} \right),{u_{ij}} = {u_{{x_i}{x_j}}}$. Let λ(x,u,p) and Λ (x,u,p) be the smallest and largest eigenvalues of the matrix ║ A_ij (z,u,p)║, so that $\lambda {\xi ^2} \leqslant {A_{ij}}{\xi _i}{\xi _j} \leqslant \Lambda {\xi ^2}$.