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Nicholas D. Alikakos

Publications and source records attributed to Nicholas D. Alikakos.

15 recordsLinked to original sources
arXiv2023

On the triple junction problem with general surface tension coefficients

We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypotheses on the solution or on the potential.

math.AP↗
arXiv2023

On the triple junction problem on the plane without symmetry hypotheses

We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution.

math.AP↗
arXiv2021

Asymptotic behavior of the free interface for entire vector minimizers in phase transitions

We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^α$ near $u=a_i$, $0<α<2$. Such solutions are, over large regions, identically equal to some zeroes of the potential $a_i$'s. We establish the estimates \begin{equation*} \mathcal{L}^n(I_0\cap B_r(x_0))\leq c_1r^{n-1},\quad \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\geq c_2r^{n-1}, \quad r\geq r_0(x_0) \end{equation*} for the diffuse interface $I_0:=\{x\in\mathbb{R}^n: \min_{1\leq i\leq N}|u(x)-a_i|>0\}$ and the free boundary $\partial I_0$. Furthermore, if $α=1$ we establish the upper bound \begin{equation*} \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\leq c_3r^{n-1}, \quad r\geq r_0(x_0). \end{equation*}

math.AP↗
arXiv2021

Entire minimizers of Allen-Cahn systems with sub-quadratic potentials

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 Euler-Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of $ W $ at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified.

math.AP↗
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↗
arXiv2014

Plateau angle conditions for the vector-valued Allen-Cahn equation

Under proper hypotheses, we rigorously derive the Plateau angle conditions at triple junctions of diffused interfaces in three dimensions, starting from the vector-valued Allen-Cahn equation with a triple-well potential. Our derivation is based on an application of the divergence theorem using the divergence-free form of the equation via an associated stress tensor.

math.AP↗
arXiv2011

A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system Δu - W_u (u) = 0

Recently, Giorgio Fusco and the author studied the system Δu - W_u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of equivariant solutions connecting the minima in certain directions at infinity, together with an estimate. In this paper a new proof is given which, in particular, avoids the introduction of a pointwise constraint in the minimization process.

math.AP↗
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↗
arXiv2010

Some basic facts on the system Δu - W_u (u) = 0

We rewrite the system Δu - W_u (u) = 0, for u: R^n to R^n, in the form div T = 0, where T is an appropriate stress-energy tensor, and derive certain a priori consequences on the solutions. In particular, we point out some differences between two paradigms: the phase-transition system, with target a finite set of points, and the Ginzburg-Landau system, with target a connected manifold.

math.AP↗