Monotonicity and stability of optimal solutions of a minimization problem

• A minimization problem related to minimum displacement of an isotropic elastic membrane is introduced.
• By deriving the minimality condition, in terms of the tangent cones, existence and uniqueness of optimal solutions are proved.
• Two monotonicity results in terms of the parameter of the admissible set are proved.
• Based on the monotonicity result we have proved the optimal solutions are stable.
• The methods employed in this paper have the potential to be applied to many other optimization problems of similar nature.

This paper is concerned with a minimization problem modeling the minimum displacement of an isotropic elastic membrane subject to a vertical force such as a load distribution. In addition to proving existence and uniqueness of optimal solutions, we show that these solutions are monotone and stable, in a certain sense. The main mathematical tool used in the analysis is the tangent cones from convex analysis, which helps to derive the optimality condition. Our results are compatible with physical expectations.