Thermodynamical and dynamical stability of Einstein-Maxwell and extremal Einstein-Born-Infeld thin shells in $(2\ \mathbf{+}\ 1)$ dimensions

We study the dynamical and thermodynamical stability of thin shells in (2+1)-dimensional spacetimes composed of an inner anti-de Sitter (AdS) region and an outer region described by a charged Ba\~nados--Teitelboim--Zanelli (BTZ) spacetime, sourced either by Einstein--Maxwell theory (Maxwell-BTZ) or Einstein--Born--Infeld theory (BI-BTZ). Assuming a fixed charge-to-mass ratio and modeling the shell's matter with a linear equation of state, we introduce a convenient parametrization to analyze the dynamical stability configurations. We find that Maxwell-BTZ thin shells admit a wider range of dynamically stable configurations compared to BI-BTZ thin shells. We also derive the thermodynamics of the shell matter, obtaining physically meaningful entropy functions in both cases, and examine the conditions for thermodynamical stability. In the Maxwell-BTZ case, we identify regions in the parameter space where configurations are both dynamically and thermodynamically stable. In contrast, for extremal BI-BTZ thin shells, all thermodynamically stable configurations are contained within the dynamically stable ones, and shells with a linear equation of state are always dynamically stable. This work extends the understanding of thin shell configurations in lower-dimensional gravity and elucidates the interplay between dynamics, thermodynamics, and nonlinear electrodynamics.