Nonlinear interpolation inequalities with fractional Sobolev norms and pattern formation in biomembranes

TL;DR

This paper introduces new nonlinear interpolation inequalities involving fractional Sobolev norms, applies them to a biological membrane model, and derives energy scaling laws relevant to pattern formation.

Contribution

The paper develops novel nonlinear interpolation inequalities that connect fractional Sobolev seminorms with Cahn-Hilliard energies, advancing mathematical tools for biological membrane modeling.

Findings

Derived scaling laws for minimal energy in membrane pattern formation

Established new bounds for fractional Sobolev seminorms using nonlinear inequalities

Applied inequalities to a variational model of biological membranes

Abstract

We consider a one-dimensional version of a variational model for pattern formation in biological membranes. The driving term in the energy is a coupling between the order parameter and the local curvature of the membrane. We derive scaling laws for the minimal energy. As a main tool we present new nonlinear interpolation inequalities that bound fractional Sobolev seminorms in terms of a Cahn-Hillard/Modica-Mortola energy.