Related papers: Equilibria and energy minimizers for an interactio…
We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the…
We consider the aggregation equation $\rho_{t}-\nabla\cdot(\rho\nabla K\ast\rho) =0$ in $\mathbb{R}^{n}$, where the interaction potential $K$ incorporates short-range Newtonian repulsion and long-range power-law attraction. We study the…
We consider the minimisation of power-law repulsive-attractive interaction energies which occur in many biological and physical situations. We show existence of global minimizers in the discrete setting and get bounds for their supports…
We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class $C^2$ and mildly repulsive at the origin; moreover, if the minimizer is global, then its…
We investigate the ground states of a free energy functional on sphere. The energy consists of an entropy and a nonlocal interaction term that are in competition with each other, as they favour spreading and aggregation, respectively.…
The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not…
We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy…
We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain…
We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances,…
In this paper we initiate the study of maps minimising the energy $$ \int_{D} (|\nabla \u|^2+2|\u|)\ dx. $$ which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations $$ \Delta…
We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive…
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially…
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with…
For the interaction energy with repulsive-attractive potentials, we give generic conditions which guarantee the radial symmetry of the local minimizers in the infinite Wasserstein distance. As a consequence, we obtain the uniqueness of…
We prove existence and uniqueness of a solution to the problem of minimizing the logarithmic energy of vector potentials associated to a $d$-tuple of positive measures supported on closed subsets of the complex plane. The assumptions we…
In this paper we study the existence of minimizers for interaction energies with the presence of external potentials. We consider a class of subharmonic interaction potentials, which include the Riesz potentials $|{\bf…
This paper deals with the global compactness and multiplicity of positive solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x) |u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N),$$ where…
We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular…
Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical…
We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modelling of aggregation phenomena. We show that in a certain regime minimizers are absolutely…