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The aim of this note is to analyse the results in the paper by D.Y. Gao and G. Strang [Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math. 47 (1989), pp. 487-504] where, according to D.Y. Gao, the complementary gap function was first introduced, and several flaws, mistakes, and inconsistencies are present, and to discuss the mathematical manner in which the mentioned paper presents itself, especially from the point of view of convex analysis. [ABSTRACT FROM AUTHOR]

The aim of this note is to show that the main results in Gao and Yang [Minimal distance between two non-convex surfaces, Optimization 57 (2008), pp. 705-714] are false and to discuss the manner in which the minimization problem in this article is investigated while presenting in a particular case a thorough study of the mentioned optimization problem. To achieve our main goal we provide counter-examples. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality ( X × X*, X* × X). [ABSTRACT FROM AUTHOR]

The aim of this article is to find explicit necessary conditions for local optimal pairs of problems governed by divergence-type elliptic PDEs, in terms depending on the nonlinearities involved in the cost functional and state equation. Several examples of optimization problems governed by ODEs and PDEs are presented. [ABSTRACT FROM AUTHOR]