This self-contained monograph presents extensions of the MoserβBangert approach that include solutions of a family of nonlinear elliptic PDEs onΒ Rn and an AllenβCahn PDE model of phase transitions. After recalling the relevant MoserβBangert results, Extensions of MoserβBangert Theory pursues the rich structure of the set of solutions of a simpler model case, expanding upon the studies of Moser and Bangert to include solutions that merely have local minimality properties. The work is intended for mathematicians who specialize in partial differential equations and may also be used as a text for a graduate topics course in PDEs Preface Contents 1. Introduction 2. Ill-posed problems 3. DSM for well-posed problems 4. DSM and linear ill-posed problems 5. Some inequalities 6. DSM for monotone operators 7. DSM for general nonlinear operator equations 8 DSM for operators satisfying a spectral assumption 9. DSM in Banach spaces 10. DSM and Newton-type methods without inversion of the derivative 11. DSM and unbounded operators 12. DSM and nonsmooth operators 13. DSM as a theoretical tool 14. DSM and iterative methods 15. Numerical problems arising in applications 16. Auxiliary results from analysis Bibliographical notes Bibliography Index