Some singularities in the behavior of solutions of equations of minimal-surface type in unbounded domains

In this paper the behavior of the solutions of equations of minimal-surface type is studied in unbounded domains. It is established that if the domain is sufficiently narrow in the neighborhood of the point at infinity of $\mathbf R^2$, then any solution having zero Dirichlet or Neumann data on the boundary must be identically constant. A condition on the narrowness of the domain is found under which the solution cannot change sign in the domain. An estimate of the form $\sum_ki(a_k)\leqslant c$ is proved, where $i(a_k)$ is the topological index of the solution at the point $a_k$, $c$ is a constant depending only on the equation, the domain and the number of points of local extremum of the boundary function, and the summation is taken over all critical points of the solution.
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