Vol. 17, No. 2, 1966

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Existence theorems for certain quasi-linear elliptic equations

Zane Clinton Motteler

Vol. 17 (1966), No. 2, 279–299

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

-∂-[a (∇φ)] = ∂ak(∇-φ)-∂2φ- = a (∇ φ)φ    = g(x,φ,∇ φ).
∂xk  k         ∂φxl  ∂xk∂xl    kl     xkxl

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.

Mathematical Subject Classification
Primary: 35.79
Received: 20 November 1964
Published: 1 May 1966
Zane Clinton Motteler