Abdelhamid Meziani
Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and have vanishing first homology group.As a consequence, a nonrigidity result for such surfaces is deduced.
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Planar loops with prescribed curvature via Hardy's inequality
本文研究具预定曲率的平面闭曲线存在性，提出基于变分法的方法，利用Hardy型不等式及相关函数空间进行证明。
We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists for all times and the network converges to the Steiner minimal connection between the three endpoints.
We study the Dirichlet problem for a graph $Σ$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $Γ=\partial Ω$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $Γ$ satisfies $h\geq H$, also suffices when $Ω$ is strictly convex, to prove the strict convexity of $Σ$.
The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus
本文研究平面平坦环面上 Mullins-Sekerka 流与保面积曲率流的平坦流解的渐近行为，利用尖锐定量 Alexandrov 不等式证明其以指数速度收敛到圆盘、条带或其补集的有限并集。
On the planar free elastic flow with small oscillation of curvature
本文研究平面自由弹性流，证明初始曲线为任意闭曲线时解全局存在，并改进曲率标量接近多重覆盖圆时的渐近稳定性结果，获得最优收敛速率。
We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of closed planar embedded curves.
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.
For $p\in (1,2]$ and a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ let $μ_p(\barΩ,\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\barΩ$ of $Ω$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $μ_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $μ$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ such that $dμ_p(\barΩ,\cdot)=dμ(\cdot)$" is solvable if and only if $μ$ has centroid at the origin and its support $\mathrm{supp}(μ)$ does not comprise any pair of …
We fix a Finsler norm $F$ and, using the anisotropic curvature flow, we prove that the anisotropic maximum curvature $k^F_{\max}$ of a smooth Jordan curve is such that $ k^F_{\max}(γ)\geq \sqrt{κ/A}$ , where $A$ is the area enclosed by $γ$ and $κ$ the area of the unitary Wulff shape associated to the anisotropy $F$.
Some aspects of anisotropic curvature flow of planar partitions
该研究聚焦平面网络的各向异性曲率流几何演化问题，分析了多光滑各向异性下经典解的局部存在性，并探讨了多晶情形的相关特性。
Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
针对平面平移平均曲率方程，通过修正近临界 grim-reaper 轮廓及长凸通道显式障碍，构造出含非凸子水平集的零狄利克雷解，为平均曲率流奇点研究提供了新实例。
Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions
研究Robin边界条件下平面带状区域中受迫曲率流的渐近行为，发现高度、速度和梯度发散，而归一化斜率与速度分别收敛于1和强迫参数，曲率测度集中于对称轴。
We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow.
In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift $γ_l(t)$ that depends on the coordinate $l$ transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of $γ$ can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allow…
IAn approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack. The classical Neumann boundary condition is augmented with curvature-dependent surface tension. It is shown that the considered model eliminates the integrable crack-tip stress and strain singularities of order 1/2 present in the classical linear fracture mechanics solutions, and also leads to the sharp crack opening that is consistent with empirical observations. Unlike for the case of a straight crack, for a general curvilinear crack some components of the stre…
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which are ``essentially'' non regular. As a first step, in this paper we study in detail the case of th…
Curvature diffusion of planar curves with generalised Neumann boundary conditions inside cones
本文研究锥内广义Neumann边界条件下四阶曲线扩散流，证明小曲率振荡保持性，并给出有限时间或全局收敛结果，推广至高阶情形。
We discuss fattening phenomenon for the evolution of planar curves according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel $K$, which is symmetric, nonnegative, possibly singular at the origin, and satisfies appropriate integrability conditions. We prove a general result about uniqueness of the geometric evolutions starting from regular sets with positive $K$-curvature and we discuss the fattening phenomenon for the evolution…
We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Under suitable regularity assumptions the $p$-elastic energy of a planar set $E\subset\mathbb{R}^2$ is defined to be $\int_{\partial E} 1 + |k_{\partial E}|^p \,\, d\mathcal{H}^1,$ where $k_{\partial E}$ is the curvature of the boundary $\partial E$. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of 1-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of $L^1$-relaxation of t…
On the generalised ideal flow of closed planar curves
研究闭嵌入平面曲线的m-理想能量，证明其临界点为圆周的多重覆盖，且在特定条件下流体渐近收敛于圆周。
We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.
Monotonicity properties of the Robin torsion function in a class of symmetric planar domains
本文证明了具有对称轴的光滑平面区域上 Robin 扭力函数在 $β\geq -κ$ 条件下的单调性，并通过构造反例证明了该条件的尖锐性。
Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions
该研究针对单一六角各向异性平面网络的晶体曲率流，证明了宽初始条件下经典解的局部存在性，分类了单有界分支的位似收缩解，并给出网络收缩为二重线段的实例。
In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.
We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.
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