\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 224, pp. 1--4.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/224\hfil Minimizer of the Ginzburg-Landau functional] {A remark on the radial minimizer of the Ginzburg-Landau functional} \author[B. Brandolini, F. Chiacchio \hfil EJDE-2014/224\hfilneg] {Barbara Brandolini, Francesco Chiacchio} % in alphabetical order \address{Barbara Brandolini \newline Dipartimento di Matematica e Applicazioni ``R. Caccioppoli'', Universit\`{a} degli Studi di Napoli ``Federico II'', Complesso Monte S. Angelo, via Cintia - 80126 Napoli, Italy} \email{brandolini@unina.it} \address{Francesco Chiacchio\newline Dipartimento di Matematica e Applicazioni ``R. Caccioppoli'', Universit\`{a} degli Studi di Napoli ``Federico II'', Complesso Monte S. Angelo, via Cintia - 80126 Napoli, Italy} \email{francesco.chiacchio@unina.it} \thanks{Submitted September 15, 2014. Published October 21, 2014.} \subjclass[2000]{35Q56, 35J15} \keywords{Ginzburg-Landau functional, Szeg\"o-Weinberger inequality} \begin{abstract} Let $\Omega\subset \mathbb{R}^2$ be a bounded domain with the same area as the unit disk $B_1$ and let $$ E_\varepsilon(u,\Omega)=\frac{1}{2}\int_\Omega |\nabla u|^2\,dx +\frac{1}{4\varepsilon^2}\int_\Omega (|u|^2-1)^2\,dx $$ be the Ginzburg-Landau functional. Denote by $\tilde u_\varepsilon$ the radial solution to the Euler equation associated to the problem $\min \{E_\varepsilon(u,B_1): \> u\big| _{\partial B_{1}}=x\}$ and by \begin{align*} \mathcal{K}=\Big\{&v=(v_1,v_2) \in H^1(\Omega;\mathbb{R}^2): \int_\Omega v_1\,dx=\int_\Omega v_2\,dx=0,\\ &\int_\Omega |v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\Big\}. \end{align*} In this note we prove that $$ \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega) \le E_\varepsilon (\tilde u_\varepsilon,B_1). $$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The Ginzburg-Landau energy has as order parameter a vectorial field \\ $u\in H^1(\Omega;\mathbb{R}^2)$ and it is defined as \begin{equation*} %\label{E} E_\varepsilon(u,\Omega)=\frac{1}{2}\int_\Omega |\nabla u|^2\,dx +\frac{1}{4\varepsilon^2}\int_\Omega \left(|u|^2-1\right)^2\,dx, \end{equation*} where $\Omega\subset\mathbb{R}^2$ is a bounded domain and $\varepsilon>0$. This kind of functionals has been originally introduced as a phenomenological phase-field type free-energy of a superconductor, near the superconducting transition, in absence of an external magnetic field. Moreover these functionals have been used in superfluids such as Helium II. In this context $u$ represents the wave function of the superflluid part of liquid and the parameter $\varepsilon$, which has the dimension of a length, depends on the material and its temperature (see \cite{GP1,GL,De}). The Ginzburg-Landau functionals have deserved a great attention by the mathematical community too. Starting from the classical monograph \cite{BBHbook} (see also \cite{BBH}) by Bethuel, Brezis and H\'elein, many mathematicians have been interested in studying minimization problems for the Ginzburg-Landau energy with several constraints, also because, besides the physical motivation, these problems appear as the simplest nontrivial examples of vector field minimization problems. In \cite{BBHbook} the authors consider Dirichlet boundary conditions $g\in C^1(\partial\Omega; \mathbb{S}^1)$ (with $\Omega$ smooth) and study the asymptotic behavior, as $\varepsilon \to 0$, of minimizers $u_\varepsilon$, which satisfy the problem \begin{equation}\label{pb} \begin{gathered} -\Delta u_\varepsilon =\frac{1}{\varepsilon^2}u_\varepsilon (1-|u_\varepsilon|^2) \quad \text{in } \Omega \\ u_\varepsilon=g \quad \text{on } \partial\Omega. \end{gathered} \end{equation} It turns out that the value $d = \deg(g,\partial\Omega)$ (i.e., the Brouwer degree or winding number of $g$ considered as a map from $\partial\Omega$ into $\mathbb{S}^1$) plays a crucial role in the asymptotic analysis of $u_\varepsilon$. In the case $\Omega=B_1$ (the unit ball in $\mathbb{R}^2$ centered at the origin), $g(x)=x$, it is natural to look for radial solutions to \eqref{pb}. Indeed, in \cite{G,BBHbook,HH} the authors prove, among other things, that \eqref{pb} has a unique radial solution, that is a solution of the form \begin{equation}\label{u} \tilde u_\varepsilon(x)=\tilde f_\varepsilon(|x|)\frac{x}{|x|} \end{equation} with $\tilde f_\varepsilon \ge 0$. Moreover $\tilde f_\varepsilon'>0$; thus, summarizing, $\tilde f_\varepsilon$ satisfies \begin{equation}\label{f} \begin{gathered} -\tilde f_\varepsilon''-\frac{\tilde f_\varepsilon'}{r} +\frac{\tilde f_\varepsilon}{r^2} =\frac{1}{\varepsilon^2}\tilde f_\varepsilon (1-\tilde f_\varepsilon^2) \quad \text{in } [0,1]\\ \tilde f_\varepsilon(0)=0,\quad \tilde f_\varepsilon(1)=1,\quad \tilde f_\varepsilon\ge 0,\quad \tilde f_\varepsilon'>0. \end{gathered} \end{equation} It is conjectured that the radial solution \eqref{u} is the unique minimizer of $E_\varepsilon$ on $B_1$. In \cite{M} (see also \cite{LL}) the author gives a partial answer to such a conjecture, proving that $\tilde u_\varepsilon$ is stable, in the sense that the quadratic form associated to $E_\varepsilon(\tilde u_\varepsilon,B_1)$ is positive definite. Other types of boundary conditions, for instance prescribed degree boundary conditions, have been considered in \cite{BR, D}. In this article we let $\Omega$ vary among domains with fixed area and prove that the map $\tilde u_\varepsilon$ in \eqref{u} provides an upper bound for the energy $E_\varepsilon$ on the class $\mathcal{K}$ we are going to introduce. \begin{theorem}\label{thm1} Let $\varepsilon>0$ and $\Omega\subset \mathbb{R}^2$ be a bounded domain such that $|\Omega|=|B_1|$. Denoted by \begin{align*} \mathcal{K}=\Big\{&v=(v_1,v_2) \in H^1(\Omega;\mathbb{R}^2): \int_\Omega v_1\,dx=\int_\Omega v_2\,dx=0,\\ & \int_\Omega |v|^2\,dx\ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx\Big\}, \end{align*} it holds \begin{equation}\label{max} \min_{v \in \mathcal{K}} E_\varepsilon (v,\Omega) \le E_\varepsilon (\tilde u_\varepsilon,B_1). \end{equation} \end{theorem} \section{Proof of Theorem \ref{thm1}} Define the following continuous extension of $\tilde f_\varepsilon$, $$ f_\varepsilon(r)=\begin{cases} \tilde f_\varepsilon(r) & \text{if } 0\le r \le 1 \\ 1 &\text{if } r>1 \end{cases} $$ and the correspondent vector field extending $\tilde u_\varepsilon$ to the whole $\mathbb{R}^2$ $$ \phi_\varepsilon(x)=\big(\phi_{\varepsilon,1}(x),\phi_{\varepsilon,2}(x)\big) =f_\varepsilon(|x|)\frac{x}{|x|}. $$ It is possible (see \cite{W}, see also \cite{AB}) to choose the origin in such a way that \begin{equation}\label{orth} \int_\Omega \phi_{\varepsilon,1}\,dx=\int_\Omega \phi_{\varepsilon,2}\,dx=0. \end{equation} Note that $\phi_\varepsilon\in \mathcal{K}$. Indeed, besides \eqref{orth}, it holds $$ \int_\Omega |\phi_\varepsilon|^2\,dx = \int_{\Omega \cap B_1} |\tilde u_\varepsilon|^2\,dx+|\Omega \setminus B_1| \ge \int_{B_1} |\tilde u_\varepsilon|^2\,dx, $$ since $|\tilde u_\varepsilon| \le 1$ in $B_1$. A direct computation yields \begin{align*} E_\varepsilon(\phi_\varepsilon,\Omega) &=\frac{1}{2}\int_\Omega \Big(f_\varepsilon'(|x|)^2 +\frac{ f_\varepsilon(|x|)^2}{|x|^2}\Big)\,dx +\frac{1}{4\varepsilon^2}\int_\Omega \big(f_\varepsilon(|x|)^2-1\big)^2\,dx\\ &=\int_\Omega B_\varepsilon(|x|)\,dx, \end{align*} where $$ B_\varepsilon(r)=\frac{1}{2}\Big(f_\varepsilon'(r)^2 +\frac{f_\varepsilon(r)^2}{r^2}\Big)+\frac{1}{4\varepsilon^2} \big(f_\varepsilon(r)^2-1\big)^2. $$ Using \eqref{f} it is straightforward to verify that \begin{equation*} %\label{decreasing} B_\varepsilon'(r)=-\frac{2}{\varepsilon^2}f_\varepsilon(r)f_\varepsilon'(r) \left(1-f_\varepsilon(r)^2\right) -\frac{1}{r}\big(f_\varepsilon'(r)-\frac{f_\varepsilon(r)}{r}\big)^2, \quad 01$, it holds $B_\varepsilon(r)=\frac{1}{2r^2}$. Thus $B_\varepsilon(r)$ is a decreasing function in $(0,+\infty)$. By Hardy-Littlewood inequality (see for instance \cite{HLP}) we finally obtain $$ E_\varepsilon(\phi_\varepsilon,\Omega) =\int_\Omega B_\varepsilon(|x|) \,dx\le \int_{B_1} B_\varepsilon(|x|)\,dx =E_\varepsilon(\tilde u_\varepsilon,B_1) $$ and hence \eqref{max}. \begin{remark} \rm The appearance of the function $\tilde u_\varepsilon$ (i.e., the candidate to be the unique minimizer of $E_\varepsilon$ in $B_1$ under the Dirichlet boundary condition $g(x)=x$) in \eqref{max} as an upper bound of the energy in the class $\mathcal{K}$ is somehow unexpected. On the other hand such a phenomenon becomes more transparent if one realizes the analogy between the problem under consideration and the maximization problem of the first nontrivial eigenvalue $\mu_1(\Omega)$ of the Neumann Laplacian among sets with prescribed area. As well-known, if $\Omega$ is a smooth, bounded domain of $\mathbb{R}^2$, $\mu_1(\Omega)$ can be variationally characterized as $$ \mu_1(\Omega)=\big\{\int_\Omega |\nabla z|^2 : z \in H^1(\Omega;\mathbb{R}),\; \int_\Omega |z|^2\,dx=1,\; \int_\Omega z \,dx=0\big\}. $$ If $|\Omega|=|B_1|$ the celebrated Szeg\"o-Weinberger inequality in the plane (see \cite{W}, see also \cite{S,B,AB,H,GP,CdB}) states \begin{equation}\label{sw} \mu_1(\Omega) \le \mu_1(B_1). \end{equation} Moreover, $\mu_1(B_1)$ is achieved by the functions $J_1(j_{1,1}'|x|)\frac{x_1}{|x|}$ or $J_1(j_{1,1}'|x|)\frac{x_2}{|x|}$, where $J_1$ is the Bessel function of the first kind and $j_{1,1}'$ is the first zero of its derivative. The role played by $J_1$ in \eqref{sw} is now played by the function $\tilde f_\varepsilon$. \end{remark} \begin{thebibliography}{99} \bibitem{AB} M. S. Ashbaugh, R. Benguria; \emph{Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature}, J. 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