Abstract
0) arises in the three- dimensional water wave theory (see M. Shinbrot [lo]), and also in the engineering problem of “hydraulic fracturing” of oil wells, see J. R. Canon and G. H. Meyer [2,7]. Mathematically, the model problem (1) is of interest, since it represents one of the simplest noncoercive elliptic problems (i.e., the Lopatinski-Shapiro condition fails, see [4]) for which there is no general theory. The field of fully nonlinear elliptic equations has recently attracted a great deal of attention (see, e.g., a recent paper of G. M. Lieberman and N. S. Trudinger [6] and the references therein to other work of the same authors as well as that of L. Evans, N. Krylov, P. L. Lions, and others). It appears that the case of second order boundary operators is not well studied, particularly for noncoercive problems. We present an existence result for (l), which extends the one in [3], where we had considered the equation
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