Related papers: Nonlocal anisotropic Riesz interactions with a phy…
The Riesz $s$-energy of an $N$-point configuration in the Euclidean space $\mathbb{R}^{p}$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to0$ the Riesz $s$-potential $1/r^s$ ($r$ the…
We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium…
We consider a large mass limit of the non-local isoperimetric problem with a repulsive Yukawa potential in two space dimensions. In this limit, the non-local term concentrates on the boundary, resulting in the existence of a critical regime…
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional…
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This…
We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies $\Gamma$-converge in the weak-$*$ topology to the Riesz functional…
We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought among all ellipses with prescribed axes or center. In the general case, we present a…
We investigate a homogenization problem related to a non-local interface energy with a periodic forcing term. We show the existence of planelike minimizers for such energy. Moreover, we prove that, under suitable assumptions on the…
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…
The purpose of this paper is threefold. First the natural extension of Riesz potentials to the context of quasi metric measure spaces for the class of upper doubling measures are studied on Lebesgue spaces, obtaining necessary and…
We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the…
This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…
We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…
In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on…
We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane. These solutions are either periodic or quasiperiodic, depending on the…
We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $\Delta_2$ nor the $\nabla _2$-condition. Fully anisotropic,…
We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density…
We study point-wise estimates for the modified Riesz potential. We show that the point-wise estimates imply embeddings into Orlicz spaces from the L^1_p-space where the functions are defined in non-smooth domains. The Orlicz functions…
We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O$(n)$ symmetric anisotropic $\phi^4$ lattice model with periodic boundary conditions in a $d$-dimensional hypercubic geometry above, at and below…
We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $\Omega\subset \>I\!\!R^{N}\>(N…