Assess whether the refined defect estimate improves strict binding

Determine whether rewriting the integral involving the exponential kernel and the symmetric-difference regions in terms of the Yukawa energy and its derivative yields a stronger strict binding inequality for the liquid drop energy with a Yukawa potential.

Background

The paper proves existence of minimizers by establishing a strict binding inequality for the minimum energy e_α(m), namely that splitting a set of volume m into two positive-volume pieces has strictly larger total energy. The proof uses scaling properties of the defect functional D_α and the isoperimetric inequality.

The authors derive a sharper comparison for D_α involving an additional integral over the symmetric difference between a set and its inner-tangent half-spaces. They observe that, if this integral could be suitably rewritten using the boundary representation, it would equal a combination of the effective surface-tension term and the derivative of α²N(E), where N is the Yukawa interaction energy. They leave unresolved whether this reformulation actually produces a stronger strict binding estimate.