This paper is devoted to
proving the existence of solutions (in the classical sense) for a certain Dirichlet
problem in the theory of quasi-linear elliptic partial differential equations of the
second order. The principal equation considered is one which can be written in the
form
If the matrix (akl) is positive definite, if the functions akl and g are Hölder
continuous in all arguments, and if the ratio of |g| to the minimum eigenvalue of (akl)
grows less rapidly than the first power of |∇φ| for large |∇φ|, then the Dirichlet
problem for φ satisfying the above equation with its values given on the sufficiently
smooth boundary of a bounded domain has a solution.
![-∂-[a (∇φ)] = ∂ak(∇-φ)-∂2φ- = a (∇ φ)φ = g(x,φ,∇ φ).
∂xk k ∂φxl ∂xk∂xl kl xkxl](a100x.png)
