On the mathematical structure and solvability of certain Hilbert space optimization problems in data-driven elasticity

In this theoretical study, we analyze the structure and solvability of data-driven elasticity problems in one spatial dimension. In contrast to Conti, Müller, Ortiz (2018, 2020), who develop an extensive, highly abstract theory for mixed Dirichlet–Neumann problems in arbitrary dimension, our setting provides a direct understanding of the problem structure and of the key issue of existence of minimizers in Hilbert space on a basic technical level. For Dirichlet problems with low regularity, we derive a reduced problem defined on orthogonal subspaces, we give explicit representations of all relevant spaces and operators, and we exploit the orthogonal decomposition to prove solvability for several standard cases and under certain symmetries. For mixed Dirichlet–Neumann problems, we prove universal solvability. In addition, we address the issue of thermomechanical consistency.