2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/193524We consider the existence and stability of static configurations of a scalar field in a five dimensional spacetime in which the extra spatial dimension is compactified on an $S^1/Z_2$ orbifold. For a wide class of potentials with multiple minima there exist a finite number of such configurations, with total number depending on the size of the orbifold interval. However, a Sturm-Liouville stability analysis demonstrates that all such configurations with nodes in the interval are unstable. Nodeless static solutions, of which there may be more than one for a given potential, are far more interesting, and we present and prove a powerful general criterion that allows a simple determination of which of these nodeless solutions are stable. We demonstrate our general results by specializing to a number of specific examples, one of which may be analyzed entirely analytically.23 pages, 7 figures, references added, factor of two corrected in kink energy definition, submitted to PRDHigh Energy Physics - PhenomenologyHigh Energy Physics - Theorytext