ترغب بنشر مسار تعليمي؟ اضغط هنا

Free boundary minimal surfaces: a nonlocal approach

103   0   0.0 ( 0 )
 نشر من قبل Alessandro Pigati
 تاريخ النشر 2017
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Given a $C^k$-smooth closed embedded manifold $mathcal Nsubset{mathbb R}^m$, with $kge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $partial S eqemptyset$, we consider $frac 12$-harmonic maps $uin H^{1/2}(partial S,mathcal N)$. These maps are critical points of the nonlocal energy begin{equation}E(f;g):=int_Sbig| ablawidetilde ubig|^2,dtext{vol}_g,end{equation} where $widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $frac 12$-energies at each boundary component of $partial S$ (suitably identified with the circle $mathcal S^1$), plus a quadratic term which is continuous in the $H^s(mathcal S^1)$ topology, for any $sinmathbb R$. We show the $C^{k-1,delta}$ regularity of $frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmuller space of $S$.



قيم البحث

اقرأ أيضاً

In this paper we establish a connection between free boundary minimal surfaces in a ball in $mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
84 - Lei Li , Mingxin Wang 2021
In this paper, we proceed to study the nonlocal diffusion problem proposed by Li and Wang [8], where the left boundary is fixed, while the right boundary is a nonlocal free boundary. We first give some accurate estimates on the longtime behavior by c onstructing lower solutions, and then investigate the limiting profiles of this problem when the expanding coefficient of free boundary converges to $0$ and $yy$, respectively. At last, we focus on two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x|^{-gamma}$ with $gammain(1,2]$ near infinity. With the help of some upper and lower solutions, we obtain some sharp estimates on the longtime behavior and rate of accelerated spreading.
We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Omegasubsetmathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly calle d chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Delta)^s$ with $sin(0,1/2)$, as defined in Fourier space. In the singular limit $varepsilonto 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Omega$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^prime$-perimeter in $Omega$ for every $s^primein(0,1/2)$.
86 - Alessandro Pigati 2020
We adapt the viscosity method introduced by Rivi`ere to the free boundary case. Namely, given a compact oriented surface $Sigma$, possibly with boundary, a closed ambient Riemannian manifold $(mathcal{M}^m,g)$ and a closed embedded submanifold $mathc al{N}^nsubsetmathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Phi$ for the functional begin{align*} &E_sigma(Phi):=operatorname{area}(Phi)+sigmaoperatorname{length}(Phi|_{partialSigma})+sigma^4int_Sigma|{mathrm {I!I}}^Phi|^4,operatorname{vol}_Phi end{align*} on immersions $Phi:Sigmatomathcal{M}$ with the constraint $Phi(partialSigma)subseteqmathcal{N}$, as $sigmato 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $mathcal{F}$ of compact subsets of the space of smooth immersions $(Sigma,partialSigma)to(mathcal{M},mathcal{N})$, assuming $mathcal{F}$ to be stable under isotopies of this space we show that the min-max value begin{align*} &beta:=inf_{Ainmathcal{F}}max_{Phiin A}operatorname{area}(Phi) end{align*} is the sum of the areas of finitely many branched minimal immersions $Phi_{(i)}:Sigma_{(i)}tomathcal{M}$ with $partial_ uPhi_{(i)}perp Tmathcal{N}$ along $partialSigma_{(i)}$, whose (connected) domains $Sigma_{(i)}$ can be different from $Sigma$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_sigma$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا