Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 743–772 | Cite as

Variational Problems with Long-Range Interaction

  • Nicola SoaveEmail author
  • Hugo Tavares
  • Susanna Terracini
  • Alessandro Zilio


We consider a class of variational problems for densities that repel each other at a distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional
$$\begin{aligned} D(\mathbf{u}) = \sum_{i=1}^k \int_{\Omega} |\nabla u_i|^2 \quad \text{or} \quad R(\mathbf{u}) = \sum_{i=1}^k \frac{\int_{\Omega} |\nabla u_i|^2}{\int_{\Omega} {u_i^2}}, \end{aligned}$$
minimized in the class of \({H^1(\Omega,\mathbb{R}^k)}\) functions attaining some boundary conditions on ∂Ω, and subjected to the constraint
$$\begin{aligned} \text{dist} (\{u_i > 0\}, \{u_j > 0\}) \ge 1 \quad \forall i \neq j. \end{aligned}$$
For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary \({\partial \{\sum_{i=1}^k u_i > 0\}}\).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.CAMGSD (Center for Mathematical Analysis, Geometry and Dynamical Systems), Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal
  3. 3.Departamento de Matemática, Faculdade de Ciências daUniversidade de LisboaLisboaPortugal
  4. 4.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTurinItaly
  5. 5.Laboratoire J.-L. Lions (CNRS UMR 7598), Université Paris Diderot - Paris 7Paris Cedex 13France

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