This paper is concerned with the asymptotic behavior of gradients of solutions of the minimal surface equation in two dimensions, for solutions which vanish on the long sides of a semi-infinite strip. Using arguments based on maximum principles, an exponential decay estimate for the gradient is obtained with a decay rate which coincides with that for Laplace's equation. The estimate is fully explicit in terms of the boundary data on the near end of the strip. The techniques are first illustrated for harmonic functions on semi-infinite strips or cylinders. The results are relevant to principles of Saint-Venant and Phragmén-Lindelöf type.
C. O. Horgan. L. E. Payne. G. A. Philippin. "Pointwise gradient decay estimates for solutions of the Laplace and minimal surface equations." Differential Integral Equations 8 (7) 1761 - 1773, 1995. https://doi.org/10.57262/die/1368397755