Related papers: Condensers with infinitely many touching Borel pla…

This work is concerned with the minimization of quantum entropies under local constraints of density, current, and energy. The problem arises in the work of Degond and Ringhofer about the derivation of quantum hydrodynamical models from…

We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence…

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results,…

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1-regularity of minimizers for the Coulombic interaction in…

An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

In this work, we study the super-Liouville equation on the sphere with positive coefficient functions. We first examine the behavior of the equation under conformal transformations and derive a Pohozaev-type identity, which generalizes the…

This paper is motivated by the study of a version of the so-called Schrodinger-Poisson-Slater problem: $$ - \Delta u + \omega u + \lambda (u^2 \star \frac{1}{|x|}) u=|u|^{p-2}u,$$ where $u \in H^1(\R^3)$. We are concerned mostly with $p \in…

This paper is the continuation of [H. Kn\"upfer and C. B. Muratov, Commun. Pure Appl. Math. (2012, to be published)]. We investigate the classical isoperimetric problem modified by an addition of a non-local repulsive term generated by a…

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$, $0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hat{\omega}^A$ of a (Radon) measure $\omega$ on $\mathbb R^n$ to a set $A\subset\mathbb…

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…

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as…

In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the…

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

We present a unified, SI-consistent framework to constrain minimal SME coefficients $a_\mu$ and $b_\mu$ using magnetically confined two-dimensional electron systems under a uniform magnetic field. Working in the nonrelativistic…

In this article, we solve the relative isoperimetric problem in $[0,1]^3$ for orthogonal polyhedra. Up to isometries of the cube or sets of measure $0$, the minimizers are of the form $[0,\epsilon]^3$, $[0,\epsilon]^2 \times [0,1]$, or…

In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces $\mathbb{R}^k$ by introducing and analyzing the notions of the $n$th constrained quantization error, constrained…

Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study…

We consider vectorial problems in the calculus of variations with an additional pointwise constraint. Our admissible mappings ${\bf n}:\mathbb{R}^k\rightarrow \mathbb{R}^d$ satisfy ${\bf n}(x)\in M$, where $M$ is a manifold embedded in…

The Poisson problem consists in finding an immersed surface $\Sigma\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a…