相关论文: Local minimizers in $3$d of vector Allen-Cahn with…

We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing…

We study vector minimizers u of the Allen-Cahn functional with potentials possessing N global minima defined on bounded domains, with certain geometrical features and Dirichlet conditions on the boundary. We derive a sharp lower bound for…

We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(\Omega)}+\int_\Omega W(u)\,dx, $$ where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the $H^s$…

We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local…

We study the minimizers of a functional on the set of partitions of a domain $\Omega \subset R^n$ into $N$ subsets $W_j$ of locally finite perimeter in $\Omega$, whose main term is $\sum_{j=1^N} \int_{\Omega \cap \partial W_j} a(x)…

We are interested in the study of local and global minimizers for an energy functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left( \mathbb{R}^N \setminus \Omega \right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy +…

We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal…

In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of $p$-Laplace Allen-Cahn energies and the second inner variation of their $\Gamma$-limit which is the area functional. Our…

The goal of this paper is to study the slow motion of solutions of the nonlocal Allen-Cahn equation in a bounded domain $\Omega \subset \mathbb{R}^n$, for $n > 1$. The initial data is assumed to be close to a configuration whose interface…

We consider a nonnegative potential $W:\mathbb{R}^2\rightarrow\mathbb{R}$ invariant under the action of the rotation group $C_N$ of the regular polygon with $N$ sides, $N\geq 3$. We assume that $W$ has $N$ nondegenerate zeros and prove the…

We establish lower semi-continuity and strict convexity of the energy functionals for a large class of vector equilibrium problems in logarithmic potential theory. This in particular implies the existence and uniqueness of a minimizer for…

In the nonlocal Almgren problem, the goal is to investigate the convexity of a minimizer under a mass constraint via a nonlocal free energy generated with some nonlocal perimeter and convex potential. In the paper, the main result is a…

We present several asymptotic results concerning the non-local Massari Problem for sets with prescribed mean curvature. In particular, we show that the fractional Massari functional $\Gamma$-converges to the classical one, and this…

This article is devoted to the study of certain models for phase transitions involving nonlocal energies. A first part is concerned with to the asymptotic analysis of a system of fractional elliptic equations of Allen-Cahn type as a…

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove…

We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a…

We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $\lambda$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $\lambda^*$, which is…

We study the constrained minimum energy problem with an external field relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant…

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the…

We investigate, under a volume constraint and among sets contained in a Euclidean half-space, the minimization problem of an energy functional given by the sum of a capillarity perimeter, a nonlocal interaction term and a gravitational…