Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such support hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.     

GLOBAL SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE HEAT-CONDUCTING FLOW WITH SYMMETRY AND FREE BOUNDARY

ABSTRACT

Global solutions of the multidimensional Navier-Stokes equations for compressible heat-conducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heat-conducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.     

Line singularity inside an elliptic inhomogeneity

The problem of an elliptic insert with a point of elastic singularity and a perfectly adhering interface is solved using the complex variable method. In particular, it is found that the remote field is insensitive to the inhomogeneity shape and interface status. Unified formulae for the special cases of free elliptic disk and rigid matrix are written and discussed. A closed-form solution for an arbitrary line singularity inside a circular inhomogeneity is also derived as a special case.     

水箱突然起动的水动压力问题

在本文中,应用Lagrange方法讨论水箱突然起动的水动压力问题.求得了自由面形状和作用在水箱壁上的动态压力系数.也证实了在自由面附近,压力系数的奇性仅是对数奇性.     

Characterization of isolated homogeneous hypersurface singularities in C4

Let V be a hypersurface with an isolated singularity at the origin in Cn+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial.For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bi.er than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Homology of homogeneous divisors

This work deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors not coming from either hyperplane arrangements or discriminants in singularity theory.     

浮体与自由面交线附近流场的奇异性

本文研究了浮体与自由面交线附近势流流场的奇异性。结果表明,线性时域解在交线附近的奇异特征是d2lnd.线性频域解在交线附近的奇异特征也是d2lnd,但若采用无穷大频率自由面条件φ=0,交线附近流场的奇异特征是d1nd,这里的d表示交线上的点与场点的距离。     

Characterization of isolated homogeneous hypersurface singularities in C~4 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let V be a hypersurface with an isolated singularity at the origin in Cn 1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Finite Gorenstein representation type implies simple singularity

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.     

Resolving singularities with Cartan’s prolongation

A standard method for resolving a plane curve singularity is the method of blow-up. We describe a less-known alternative method which we call prolongation, in honor of Cartan’s work in this direction. This method is known to algebraic geometers as Nash blow-up. With each application of prolongation the dimension of the ambient space containing the new “prolonged” singularity increases by one. The new singularity is tangent to a canonical plane field on the ambient space. Our main result asserts that the two methods, blow-up and prolongation, yield the same resolution for unibranched singularities. The primary difficulties encountered are around understanding the prolongation analogues of the exceptional divisors from blow-up. These analogues are called critical curves. Most of the critical curves are abnormal extremals in the sense of optimal control theory as it applies to rank 2 distributions (2 controls). Dedicated to V. I. Arnol’d and his creative force     

On Regularity and Singularity of Free Boundaries in Obstacle Problems

The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems. This approach is based on the monotonicity of several variational integrals, the Federer-Almgren dimension reduction and stratification theorems, and some simple PDE arguments.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mean curvature flow with free boundary – Type 2 singularities

In this paper we give sufficient conditions that guarantee the mean curvature flow with free boundary on a pinched cylinder develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.     

Finite time singularity in a free boundary problem modeling MEMS

The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term taking into account large deformations.     

一个新的有自由面渗流问题的变分不等式提法

建立了一个新的有自由面渗流问题的变分不等式提法,提法通过将潜在出渗面上的边界条件提为Signorini型条件,从而从理论上消除了出渗点的奇性并解决了出渗点的定位问题．与其它变分不等式提法相比,该提法有更好的数值稳定性．     

Limit holonomy and extension properties of Sobolev and Yang-Mills bundles

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally L^m/2-integrable curvature and trivial limit holonomy extends through singularity of codimension 2. Additionally, if the connection satisfies the Yang-Mills-Higgs equation, the extension also satisfies the equation.     

球壳和柱壳振动中一类方程组的求解

本文利用矩阵型式的Ｆｒｏｂｅｎｉｕｓ级数方法求解了球壳和柱壳振动中一类含正则奇点的常微分方程组，考虑了指标方程根的相互关系，从而全面地得到了不同情况下解的表达形式，为解析求解工作奠定了基础。     

Partial regularity for a minimum problem with free boundary

Regularity of the free boundary ?{u > 0} of a non-negative minimum u of the functional $\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)} $ , where Ω is an open set in ?ⁿ and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3. By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ?{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets. This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ?{u > 0} is locally in Ω a C^1,α-surface, for n = k* the singular set Σ:= ?{u > 0} ? ?_red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.     

The uniqueness of tangent cones for Yang–Mills connections with isolated singularities

We proved a uniqueness theorem of tangent connections for a Yang–Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained control over the rate of the asymptotic convergence of the connection to the tangent connection if furthermore the connection is stationary or the tangent connection is integrable, with a stronger result in the latter case. There are parallel results for the cones at infinity of a Yang–Mills connection on an asymptotically flat manifold. We also gave an application of our methods to the Yang–Mills flow and proved that the Yang–Mills flow exists for all time and has asymptotic limit if the initial value is close to a smooth local minimizer of the Yang–Mills functional.