Ihsan Topaloglu

Existence of Ground States of Nonlocal-Interaction Energies

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e.

Sharp interface limit of an energy modelling nanoparticle-polymer blends

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On minimizers of interaction functionals with competing attractive and repulsive potentials

H-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.

On a nonlocal isoperimetric problem on the two-sphere

Inspired by numerical studies of the aggregation equation, we study the effect of regularization on nonlocal interaction energies. We consider energies defined via a repulsive-attractive interaction kernel, regularized by convolution with a mollifier. We prove that, with respect to the 2-Wasserstein metric, the regularized energies

An Old Problem Resurfaces Nonlocally: Gamow's Liquid Drops Inspire Today's Research and Applications

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Nonlocal Shape Optimization via Interactions of Attractive and Repulsive Potentials

-converge to the unregularized energy and minimizers converge to minimizers. We then apply our results to prove

Existence of Minimizers of Nonlocal Interaction Energies

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Ground-states for the liquid drop and TFDW models with long-range attraction

-convergence of the gradient flows, when restricted to the space of measures with bounded density.