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

Abstract 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 Bañados–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.