Preservation of Quasiconvexity and Quasimonotonicity in Polynomial Approximation of Variational Problems

In this thesis, we are concerned with three classes of non-linear problems that appear naturally in various fields of science, engineering and economics. In order to cover many different applications, we study problems in the calculus of variation (Chapter 3), partial differential equations (Chapter 4) as well as non-linear programming problems (Chapter 5). As an example of possible applications, we consider models of non-linear elasticity theory. The aim of this thesis is to approximate a given non-linear problem by polynomial problems. In other words: A given non-linear problem is associated with a number of non-linear functions that serve as parameters and represent the non-linear problem. We show that these non-linear functions can be approximated by algebraic polynomials so that characteristic properties of the corresponding problems are preserved. Polynomial approximation is interesting, since tools that can be applied to polynomial problems are not available for non-polynomial (non-linear) problems in general. In order to achieve the desired polynomial approximation of problems, a large part of this thesis is dedicated to the polynomial approximation of nonlinear functions. The Weierstraß approximation theorem forms the starting point. Based on this well-known theorem, we prove theorems that eventually lead to our main result: A given non-linear function can be approximated by polynomials so that essential properties of the function are preserved. This result is new for three properties that are important in the context of the considered non-linear problems. These properties are: quasiconvexity in the sense of the calculus of variation, quasimonotonicity in the context of partial differential equations and quasiconvexity in the sense of non-linear programming (Theorems 3.16, 4.10 and 5.5). Several theorems in this thesis deal with polynomial approximation of non-linear functions on an abstract level (Theorems 3.7 and 4.5). The abstract approach is useful, since its results can be applied to various notions of convexity and of monotonicity. Moreover, it forms the basis for the polynomial approximation of non-linear problems. Finally, we show the following: Every non-linear problem that belongs to one of the three considered classes of problems can be approximated by polynomial problems (Theorems 3.26, 4.16 and 5.8). The underlying convergence guarantees both the approximation in the parameter space and the approximation in the solution space. In this context, we use the concepts of Gamma-convergence (epi-convergence) and of G-convergence.