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Giorgio Fusco

Publications and source records attributed to Giorgio Fusco.

17 recordsLinked to original sources
arXiv2024

Connectivity of the diffuse interface and fine structure of minimizers in the Allen-Cahn theory of phase transitions

In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny Diffuse Interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers. We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface. Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well defined structure. This network is contained in the diffuse interface and is a priori unknown. Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume that the phase are connected, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular we can characterize the shape and the size of the various phases and also how they depend on the surface tensions.

math.AP↗
arXiv2023

On the structure of the infinitesimal generators of semigroups with discrete Lyapunov functionals

Dynamical systems generated by scalar reaction-diffusion equations on an interval enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system. Other systems, like the linear scalar delay differential equations under monotone feedback conditions, possess similar kinds of discrete Lyapunov functions. Here we discuss and characterize classes of linear equations that generate semiflows acting on $C^0[0,1]$ or on $C^1[0,1]$ which admit discrete Lyapunov functions related to the zero number. We show that, if the space is $C^1[0,1]$, the corresponding equations are essentially parabolic partial differential equations. In contrast, if the space is $C^0[0,1]$, the corresponding equations are generalizations of monotone feedback delay differential equations.

math.DS↗
arXiv2021

Sharp lower bounds for vector Allen-Cahn energy and qualitative properties of minimizes under no symmetry hypotheses

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 the energy (as ε{\rightarrow} 0) with the additional feature that it involves half of the gradient and part of the domain. Based on this we derive very precise (in ε) pointwise estimates up to the boundary for uε. Depending on the geometry of the domain uε exhibits either boundary layers or internal layers. We do not impose symmetry hypotheses.

math.AP↗
arXiv2021

Minimizing under relaxed symmetry constraints: Triple and $N$-junctions

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 existence of a $N$-junction solution to the vector Allen-Cahn equation. The proof is variational and is based on sharp lower and upper bounds for the energy and on a new pointwise estimate for vector minimizers.

math.AP↗
arXiv2020

Periodic motions for multi-wells potentials and layers dynamic for the vector Allen-Cahn equation

We consider a nonnegative potential $W$ that vanishes on a finite set and study the existence of periodic orbits of the equation \[\ddot{u}=W_u(u),\;\;t\in\R,\] that have the property of visiting neighborhoods of zeros of $W$ in a given finite sequence. We give conditions for the existence of such orbits. After introducing the new variable $x=εt$, $ε>0$ small, these orbits correspond to stationary solutions of the parabolic equation \[u_t=u_{xx}-W_u(u),\;\;x\in(0,1),\;t>0,\] with periodic boundary conditions. In the second paper of the paper we study solutions of this equation that, as the stationary solutions, have a layered structure. We derive a system of ODE that describes the dynamics of the layers and show that their motion is extremely slow.

math.AP↗
arXiv2018

Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} Δu=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional \[J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.\] We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.

math.DS↗
arXiv2017

On the existence of connecting orbits for critical values of the energy

We consider an open connected set $Ω$ and a smooth potential $U$ which is positive in $Ω$ and vanishes on $\partialΩ$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different components of $\partialΩ$ and lie on the zero level of the energy. We allow that $\partialΩ$ contains a finite number of critical points of $U$. The case of symmetric potential is also considered.

math.DS↗
arXiv2016

Layered solutions to the vector Allen-Cahn equation in $ R^2$. Characterization of minimizers and a new approach to heteroclinic connections

Let $W:R^m\rightarrow R$ be a nonnegative potential with exactly two nondegenerate zeros $a_-\neq a_+\in R^m$. We assume that there are$ N\geq 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $ u_1,\ldots,u_N$ that minimize the one-dimensional energy $J_R(u) =\int_R(\frac{\vert u^\prime\vert^2}{2}+W(u))ds$. We first consider the problem of characterizing the minimizers $u:R^n\rightarrow R^m$ of the energy $\mathcal{J}_Ω(u) =\int_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u))dx$. Under a nondegeneracy condition on $ u_1,\ldots,u_N $ and in two space dimensions, we prove that, provided it remains away from $a_-$ and $a_+$ in corresponding half spaces $S_-$ and $S_+$, a bounded minimizer $u:R^n\rightarrow R^m$ is necessarily an heteroclinic connection between suitable translates $ u_-(. -η_-)$ and $ u_+(. -η_+) $ of some $ u_\pm\in\{ u_1,\ldots, u_N\}$. Then we focus on the existence problem and assuming $N = 2$ and denoting $ u_-$ and $ u_+$ the representations of the two orbits connecting $ a_-$ to $ a_+$ we give a new proof of the existence (first proved in [31]) of a solution $ u:R^2\rightarrow R^m $ of \[Δu = W_u(u),\] that connects certain translates of $ u_\pm $.

math.AP↗
arXiv2014

Multyphase solutions to the vector Allen-Cahn equation: Crystalline and other complex symmetric structures

We present a systematic study of entire symmetric solutions $u:R^n\rightarrow R^m$ of the vector Allen-Cahn equation $Δu-W_u(u)=0, x \in R^n$, where $W:R^m\rightarrow R$ is smooth, symmetric, nonnegative with a finite number of zeros and $W_u=(\frac{\partial W}{\partial u_1},\ldots,\frac{\partial W}{\partial u_m})^\top$. We introduce a general notion of equivariance with respect to a homomorphism $f:G\rightarrowΓ$ ($G,Γ$ reflection groups) and prove two abstract results, concerning the cases of $G$ finite and $G$ discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen-Cahn equation and on a pointwise estimate for vector minimizers.

math.AP↗
arXiv2014

On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in ${\bf R}^2$

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension n=2 an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.

math.AP↗
arXiv2012

Stationary motion of a self gravitating toroidal incompressible liquid layer

We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal tickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radii r, R and R>>r) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius R that connects the centers of the cross sections of the torus.

math-ph↗
arXiv2011

Entire solutions to equivariant elliptic systems with variational structure

In the present paper we consider the system Δu - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.

math.AP↗
arXiv2010

On an elliptic system with symmetric potential possessing two global minima

We consider the system Δu - W_u (u) = 0, for u: R^2 -> R^2, W: R^2 -> R, where W_u (u) is a smooth potential, symmetric with respect to the u_1, u_2 axes, possessing two global minima a^\pm := (\pma,0) and two connections e^\pm(x_1) connecting the minima. We prove that there exists an equivariant solution u(x_1, x_2) satisfying u(x_1, x_2) -> a^\pm, as x_1 -> \pminfiniti, and u(x_1, x_2) -> e^\pm(x_1), as x_2 -> \pminfiniti. The problem above was first studied by Alama, Bronsard, and Gui under related hypotheses to the ones introduced in the present paper. At the expense of one extra symmetry assumption, we avoid their considerations with the normalized energy and strengthen their result. We also provide examples for W.

math.AP↗