This dissertation is concerned with some of the many applications of the MAXIMUM PRINCIPLE. In the first two chapters, we discuss and prove versions of the maximum principle first for Ordinary Differential equations, then for elliptic Partial Differential Equations, including some improvements due to Serrin. In Chapter (III), we study in detail symmetry properties of positive solutions of second order elliptic equations of the type Delta u+ f(u) = 0 in a domain O with zero boundary conditions. This follows the important article of Gidas, Ni and Nirenberg and shows that the problem cited has radial solutions in a spherically symmetric domain, no matter what the function f is. We give extensions of these results to certain systems of second order elliptic equations in Chapter (IV). Chapters (V) and (VI) contain applications of different type. In Chapter (V), we study solutions of the equation Delta u+ f(u) = 0 with either Dirichlet or Neumann boundary conditions, and obtain bounds for various quantities determined by a solution of Delta u+ f(u) = 0. We show that it is possible to find functions g, f so that the function P = g(u) |grad u|2 + h(u) satisfies an elliptic inequality and , by an application of the maximum principle, P either attains its maximum on the boundary of O, or at a critical point of u. We study particularly the case h'(u) = c f(u) g(u) where c is a constant. For c ≤ 1 we show that, under suitable assumptions, the maximum of P occurs on ∂O, whereas for c ≥ 2 the maximum occurs at a critical point of u. In the last chapter, we illustrate these results by giving some applications to the torsion problem, the efficiency ratio of a nuclear reactor and the free membrane problem
