ABSTRACT
The scope of this paper it to review a number of possible applications of the maximum principle. The second section of the paper deals with a free boundary Stekloff eigenvalue problem. The third section is concerned with gradient estimates for harmonic functions defined in a convex region Ω ⊂ ℝ2 in terms of boundary data. Our estimates turn out to be valid in more general cases, such as for classical solutions of the minimal surface equation. Similar methods of investigation are then applied in the last section of the paper to obtain pointwise gradient decay estimates for solution of the Laplace equation.
