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Isolated domains in lipid monolayers exhibit shape transitions as aconsequence of the competing effects of line tension and long-range dipolarinteractions. Nontrivial integrals ultimately arise in instability analysiscalculations of these domains. In this work we present a closed form evaluationof such integrals in terms of associated Legendre functions. Our approach ismotivated by and extends the work of J. M. Deutch and F. E. Low (J. Phys.Chem., 1992, 96, 7097-7101). Our closed form solutions lead to simpler analyticexpressions for stability thresholds, which are easier to evaluate explicitly.Relevant asymptotic behavior is calculated in various limits of interest. Forthe case of Coulomb interactions, a general closed form expression is obtainedfor the critical domain radii.

Closed Form Results for Shape Transitions in Lipid Monolayer Domains