Classification of the pinned solution branch and resolution of transition thresholds

Determine whether the pinned solution branch in Example 3 is a second strict local minimum or a saddle, and resolve the parameter values marking the edges of the interval in which the two solution branches coexist.

Background

For Example 3, the numerical computations identify two solution branches of the same discrete problem over a finite interval of α\alpha values. Both computed solutions are reported to be strict local minimizers based on positive definiteness of the reduced Hessian, but the authors explicitly state that the precise variational classification of the pinned branch has not been determined. The transition values at which the active branch ceases to be globally minimizing and later ceases to exist are also only bracketed by the sampled parameter grid, leaving their exact locations unresolved.

References

We have not determined whether the pinned branch is a second local minimum or a saddle, and the edges of the window are bracketed by the grid of Figure~\ref{fig:branches} rather than resolved.

— Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization  (2609.35538 - Bartman-Szwarc et al., 28 Sep 2026) in Section 4, Numerical Results, paragraph following Figure \ref{fig:temperature}