Partial Differential Equations and Inverse Problems in Non-Separable Banach Spaces
Abstract
Juan Alberto Molina Garcia
The classical functional–analytic theory of partial differential equations (PDEs) is predominantly developed within separable Banach and Hilbert spaces, where compactness, countable bases, and metrisation techniques play a central methodological role. However, a growing number of mathematical models arising in infinite-dimensional dynamics, continuum mechanics, control theory, inverse problems, and data-driven systems naturally lead to non-separable functional settings. This article develops a systematic framework for the analysis of partial differential equations and inverse problems in NSBS, extending existence, uniqueness, and stability results beyond the separable paradigm.
We introduce a functional-analytic formulation of PDEs on NSBS that avoids reliance on compact embeddings or countable approximations, replacing them with weak, weak* and approximate compactness structures adapted to non-separable contexts. Constructive examples are provided in canonical spaces such as l∞ (Γ) and C(K) for non- metrisable compact spaces K, illustrating how classical PDE phenomena persist—or fundamentally change—under loss of separability.
A particular emphasis is placed on inverse problems for nonlinear dynamical systems, where the unknown parameters, operators, or forcing terms naturally inhabit non-separable spaces. We propose an abstract inverse-problem framework in EBNS, combining approximate solvability, weak stability, and operator-theoretic regularisation. Within this setting, adapted existence and uniqueness theorems are established, highlighting the precise functional-analytic conditions under which identifiability and reconstruction remain valid.
The results presented contribute to a broader programme aimed at extending PDE theory, inverse problems, and applied analysis to non-separable environments, offering a mathematically rigorous foundation for modelling real-world systems whose intrinsic complexity exceeds separable functional representations.