arXiv subjects
Publications and source records attributed to Petru Mironescu.
We develop a set of tools for the asymptotic analysis of minimizers of the anisotropic Ginzburg-Landau functional among the admissible competitors with Dirichlet boundary datum of negative degree.
We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{N}) \simeq \{0\}$. We present several open problems connected to the extension problem.
Let $π: \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, $0 1$, in dimension $1$. Our argument also leads to the existence of a lifting when $0<s<1$ and $1<sp<2\le m$, provided there is no topological obstruction on $u$, i.e., $u = π\, \circ \, \widetilde{u}$ holds in this range provided $u$ is in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$. However, when $0<s<1$, $sp = 1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}(\mathcal{M},\mathcal{N})$ in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$, such that $u = π\, \circ \, \widetilde{u}$ does not hold for any $\widetilde{u} \in W^{s, p} ({\mathcal{M}}, \widetilde{\mathcal{N}} )$.
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 $W^{s,p}(B)$.
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 $p>1$.
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.
Let $Ω$ be a smooth bounded domain in $\mathbb R^n$ and u be a measurable function on $Ω$ such that $|u(x)|=1$ almost everywhere in $Ω$. Assume that u belongs to the $B^s_{p,q}(Ω)$ Besov space. We investigate whether there exists a real-valued function $φ\in B^s_{p,q}$ such that $u=e^{iφ}$. This extends the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The analysis of this lifting problem leads us to prove some interesting new properties of Besov spaces, in particular a non restriction property when $q>p$.
We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special cases we even obtain exact formulas.
Let $Ω$ be a smooth bounded domain in ${\mathbb R}^n$, $0\textless{}s\textless{}\infty$ and $1\le p\textless{}\infty$. We prove that $C^\infty(\overlineΩ\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(Ω; {\mathbb S}^1)$ except when $1\le sp\textless{}2$ and $n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when $s\textless{}1$. With $0\textless{}s\textless{}1$, $1\le p\textless{}\infty$ and $sp\textless{}n$, $Ω$ a ball, and $N$ a general compact connected manifold, we prove that $C^\infty(\overlineΩ\, ; N)$ is dense in $W^{s,p}(Ω\, ; N)$ if and only if $π\_{[sp]}(N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\ldots$ [General domains $Ω$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to $s\textless{}1$.] The case where $s\textgreater{}1$, $s\not\in{\mathbb N}$, is still open.
Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\astρ_\varepsilon$ to $f$, where $ρ$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $ρ$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $ρ$ is a characteristic function.
Let $Ω$ be a smooth bounded simply connected domain in $\mathbb{R}^2$. We investigate the existence of critical points of the energy $E_\varepsilon (u)=1/2\int_Ω|\nabla u|^2+1/(4\varepsilon^2)\int_Ω(1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $Ω$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in "most" of the domains.
We study the structure of vortex solutions in a Ginzburg-Landau system for two complex valued order parameters. We consider the Dirichlet problem in the disk in R^2 with symmetric, degree-one boundary condition, as well as the associated degree-one entire solutions in all of R^2. Each problem has degree-one equivariant solutions with radially symmetric profile vanishing at the origin, of the same form as the unique (complex scalar) Ginzburg-Landau minimizer. We find that there is a range of parameters for which these equivariant solutions are the unique locally energy minimizing solutions for the coupled system. Surprisingly, there is also a parameter regime in which the equivariant solutions are unstable, and minimizers must vanish separately in each component of the order parameter.