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Itai Shafrir

Publications and source records attributed to Itai Shafrir.

14 recordsLinked to original sources
arXiv2025

A mixed eigenvalue problem on domains tending to infinity in several directions

The aim of this article is to analyze the asymptotic behaviour of the eigenvalues of elliptic operators in divergence form with mixed boundary type conditions for domains that become unbounded in several directions, while they stay bounded in some directions (cylindrical domains). The limiting behavior of such eigenvalues is shown to depend on an ensemble of eigenvalue problems defined on a domain that is unbounded only in one direction. The asymptotic behavior of the eigenfunctions are also discussed. This work is a continuation of the work done in [6].

math.AP↗
arXiv2021

A variational singular perturbation problem motivated by Ericksen's model for nematic liquid crystals

We study the asymptotic behavior, when $\varepsilon\to0$, of the minimizers $\{u_\varepsilon\}_{\varepsilon>0}$ for the energy \begin{equation*} E_\varepsilon(u)=\int_Ω\Big(|\nabla u|^2+\big(\frac{1}{\varepsilon^2}-1\big)|\nabla|u||^2\Big), \end{equation*} over the class of maps $u\in H^1(Ω,{\mathbb R}^2)$ satisfying the boundary condition $u=g$ on $\partialΩ$, where $Ω$ is a smooth, bounded and simply connected domain in ${\mathbb R}^2$ and $g:\partialΩ\to S^1$ is a smooth boundary data of degree $D\ge1$. The motivation comes from a simplified version of the Ericksen model for nematic liquid crystals with variable degree of orientation. We prove convergence (up to a subsequence) of $\{u_\varepsilon\}$ towards a singular $S^1$-valued harmonic map $u_*$, a result that resembles the one obtained in \cite{BBH} for an analogous problem for the Ginzburg-Landau energy. There are however two striking differences between our result and the one involving the Ginzburg-Landau energy. First, in our problem the singular limit $u_*$ may have singularities of degree strictly larger than one. Second, we find that the principle of \enquote{equi-partition} holds for the energy of the minimizers, i.e., the contributions of the two terms in $E_\varepsilon(u_\varepsilon)$ are essentially equal.

math.AP↗
arXiv2019

Existence of superconducting solutions for a reduced Ginzburg-Landau model in the presence of strong electric currents

In this work we consider a reduced Ginzburg-Landau model in which the magnetic field is neglected and the magnitude of the current density is significantly stronger than that considered in a recent work by the same authors. We prove the existence of a solution which can be obtained by solving a non-convex minimization problem away from the boundary of the domain. Near the boundary, we show that this solution is essentially one-dimensional. We also establish some linear stability results for a simplified, one-dimensional version of the original problem.

math-ph↗
arXiv2018

Best constants for two families of higher order critical Sobolev embeddings

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into $L^\infty(\mathbb{R}^N)$ and those that embed into slightly larger target spaces. Concerning the former, we show that for $k \in \{1,\ldots, N-1\}$, $N-k$ even, one has an optimal constant $c_k>0$ such that \[ \|u\|_{L^\infty} \leq c_k \int |\nabla^k (-Δ)^{(N-k)/2} u|\] for all $u \in C^\infty_c(\mathbb{R}^N)$ (the case $k=N$ was handled in a recent paper by Shafrir). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For $Ω\subset \mathbb{R}^N$, $m \in \mathbb{N}$ and $p=\frac{N}{m}$, there exists $A>0$ and optimal constant $β_0>0$ such that \[ \int_Ω \exp (β_0 |u|^{p^\prime}) \leq A |Ω| \] for all $u$ such that $\|\nabla^m u\|_{L^p(Ω)} \leq 1$, where $\|\nabla^m u\|_{L^p(Ω)}$ is the traditional semi-norm on the space $W^{m,p}(Ω)$.

math.AP↗
arXiv2018

Radial extensions in fractional Sobolev spaces

Given $f:\partial (-1,1)^n\to{\mathbb R}$, consider its radial extension $Tf(X):=f(X/\|X\|_{\infty})$, $\forall\, X\in [-1,1]^n\setminus\{0\}$. In "On some questions of topology for $S^1$-valued fractional Sobolev spaces" (RACSAM 2001), the first two authors (HB and PM) stated the following auxiliary result (Lemma D.1). If $0 0$, $1\le p<\infty$ and $n\ge 2$ be such that $(s-a)p<n$. Then $f\mapsto U_af$ is a bounded linear operator from $W^{s,p}(\partial B)$ into $W^{s,p}(B)$.

math.FA↗
arXiv2017

On the distance between homotopy classes in $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)$

For every $p\in(1,\infty)$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\mathbb S}^1;{\mathbb S}^1)=\bigcup_{d\in{\mathbb Z}}\mathcal{E}_d$. For every pair $d_1,d_2\in {\mathbb Z}$, we show that the distance $\text{Dist}_{W^{1/p,p}}({\mathcal E}_{d_1}, {\mathcal E}_{d_2}):=\sup_{f\in{\mathcal E}_{d_1}}\ \inf_{g\in{\mathcal E}_{d_2}}\ d_{W^{1/p,p}}(f, g)$ equals the minimal $W^{1/p,p}$-energy in $\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce from the latter formula an explicit value: $\text{Dist}_{W^{1/2,2}}({\mathcal E}_{d_1}, {\mathcal E}_{d_2})=2π|d_2-d_1|^{1/2}$.

math.FA↗
arXiv2017

Distances between classes in $W^{1,1}(Ω;{\mathbb S}^1)$

We introduce an equivalence relation on the space $W^{1,1}(Ω;{\mathbb S}^1)$ which classifies maps according to their "topological singularities". We establish sharp bounds for the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. Similar questions are examined for the space $W^{1,p}(Ω;{\mathbb S}^1)$ when $p>1$.

math.FA↗
arXiv2017

Asymptotic behavior of critical points of an energy involving a loop-well potential

We describe the asymptotic behavior of critical points of $\int_Ω [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $Γ$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and Hélein, we do not assume any symmetry on $W$ or $Γ$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest.

math.AP↗
arXiv2017

Small energy Ginzburg-Landau minimizers in ${\mathbb R}^3$

We prove that a local minimizer of the Ginzburg-Landau energy in ${\mathbb R}^3$ satisfying the condition $\liminf_{R\to\infty}E(u;B_R)/RlnR < 2π$ must be constant. The main tool is a new sharp eta-ellipticity result for minimizers in dimension three that might be of independent interest.

math.AP↗
arXiv2010

Radially symmetric minimizers for a $p$-Ginzburg Landau type energy in $\R^2$

We consider the minimization of a p-Ginzburg-Landau energy functional over the class of radially symmetric functions of degree one. We prove the existence of a unique minimizer in this class, and show that its modulus is monotone increasing and concave. We also study the asymptotic limit of the minimizers as p \rightarrow \infty. Finally, we prove that the radially symmetric solution is locally stable for $p$ in the interval $(2,4]$.

math.AP↗