positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane abstract we describe the evolution under the mean curvature flow of embedded lagrangian spherical surfaces in the complex euclidean plane c 2 in particular we answer the question 4 7 addressed in 10 by a neves about finding out a condition on a starting lagrangian torus in c 2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the clifford torus

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

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

ancient solutions to mean curvature flow for isoparametric submanifolds mean curvature flow for isoparametric submanifolds in euclidean spaces and spheres was studied in liu and terng duke math j 147 1 157 179 2009 in this paper we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity we also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the chern s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres

convexity and horizontal second fundamental forms for hypersurfaces in carnot groups we use a riemannian approximation scheme to give a characterization for smooth convex functions on a carnot group in the sense of danielli garofalo nhieu or lu manfredi stroffolini in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

mean curvature flow of entire graphs in a half space with a free boundary we study the mean curvature flow of graphs with prescribed contact angle on a fixed smooth hyperplane in euclidean space we obtain long time existence and convergence to a self similar solution of the mean curvature flow orthogonal to the fixed hyperplane

spectral factorization by optimal gain iteration an iterative algorithm for spectral factorization is presented the algorithm has the form of a frequency domain version of the newton raphson solution of the corresponding riccati equation although results are established without reference to an underlying state variable model application is made to a distributed filtering problem

a newton raphson method for moving average spectral factorization using the euclid algorithm an implementation of the newton raphson approach to compute the minimum phase moving average spectral factor of a finite positive definite correlation sequence is presented each step in the successive approximation method involves a system of linear equations that is solved using either the levinson algorithm backwards the jury stability test or a symmetrized version of the euclid algorithm various properties of the newton raphson map are studied the algorithm is generalized to other symmetries other than with respect to the unit circle the special case of the symmetry with respect to the imaginary axis is presented and related to the routh hurwitz stability test for continuous time transfer function

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

the asymptotic behavior of symplectic mean curvature flow with pinched curvatures in cp2 abstract it was proved by han li yang 4 that the mean curvature flow of symplectic surfaces with suitably pinched curvatures in c p 2 has a longtime solution and converges to a holomorphic curve as the time approaches infinity in this note we give a refinement of this theorem we prove that the evolving surface becomes more and more umbilical and the limit surface is in fact c p 1 as a consequence the initial surface is symplectically isotopic to c p 1

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

manifolds with positive curvature operators are space forms the ricci flow was introduced by hamilton in 1982 h1 in order to prove that a compact three manifold admitting a riemannian metric of positive ricci curvature is a spherical space form in dimension four hamilton showed that compact four manifolds with positive curvature operators are spherical space forms as well h2 more generally the same conclusion holds for compact four manifolds with 2 positive curvature operators che recall that a curvature operator is called 2 positive if the sum of its two smallest eigenvalues is positive in arbitrary dimensions huisken hu described an explicit open cone in the space of curvature operators such that the normalized ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature hamilton conjectured that in all dimensions compact riemannian manifolds with positive curvature operators must be space forms in this paper we confirm this conjecture more generally we show the following

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions we study the bifurcation of radially symmetric solutions of f u 0 onn balls into asymmetric ones we show that ifu satisfies homogeneous neumann boundary conditions the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension for general boundary conditions we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions and on the structure of the kernels of the linearized operators

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

asymptotic behavior of flows by powers of the gaussian curvature we consider a 1 parameter family of strictly convex hypersurfaces in mathbb r n 1 moving with speed k alpha where denotes the outward pointing unit normal vector and alpha geqslant 1 n 2 for alpha gt 1 n 2 we show that the flow converges to a round sphere after rescaling in the affine invariant case alpha 1 n 2 our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling

evolution of geometric statistics a model for the description of the geometry in multiphase mechanics is derived and studied equations for the rate of change of mean and gauss curvature are derived these coupled equations show that the evolution of these curvatures depend on the velocity of the interface normal to itself examples are given to show the utility of these equations to apply these concepts to multiphase mechanical situations an ensemble average is applied to the curvature equations equations for the evolution of the volume fraction and the interfacial area density are also derived these equations also depend on the curvatures and the average interfacial velocity examples are given to show that these equations give a description of the geometry of multiphase mechanics that is superior to the present description which describes the force between materials as depending on two geometric parameters the volume fraction and the average radius of the dispersed material these examples also suggest constitutive equations to d

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

a low dimensional conjugacy for elliptic equations and symmetry breaking on rotated domains abstract a topological conjugacy is established between certain elliptic pdes with one unbounded time direction and a simple second order differential equation admitting the dynamics of such pdes to be examined on a two dimensional submanifold by this means periodic solutions can be obtained to elliptic equations as perturbations of those that are independent of time

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

continuum modeling of flexible structures with application to vibration control in this paper we consider the construction of hybrid models for flexible structures including both the natural spatially distributed dynamics of low mass density structures and the lumped models associated with rigid body dynamics for the purposes of active control of structural vibrations we argue that by using transform methods the standard approach to transfer function or mechanical impedance modeling can be extended to provide a complete formulation of a hybrid model with localized control we proceed to give a systematic procedure for obtaining the required transfer functions and green s functions for the hybrid model dynamics consisting of one dimensional elastic elements such as beams rods and cables together with rigid body lumped models we focus on a well known family of models for elastic elements together with various reductions and indicate the construction of well posed state space models for these systems using the resulting state space we next demonstrate the construction of a complete frequency domain characterization of the elemental system response using transfer functions and green s functions hybrid constructions are discussed for various interconnections of distributed and lumped elements and a simple illustrative example is given we highlight the extent to which the required computations can be performed automatically on a digital computer using a computer algebra system

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

the extension and convergence of mean curvature flow in higher codimension in this paper we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a riemannian manifold with codimension d 1 which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to le ysum 12 and the authors 25 26 using the extension theorem we prove two convergence theorems for the mean curvature flow of closed submanifolds in r n d under suitable integral curvature conditions

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

deforming hypersurfaces of the sphere by their mean curvature is satisfied in 6 we studied hypersurfaces moving along their mean curvature vector in a general riemannian manifold n 1 it was shown that all hypersurfaces mo satisfying a suitable convexity condition will contract to a single point in finite time during this evolution here we want to show that in a spherical spaceform some convergence results can be obtained without assuming convexity for the initial hypersurface mo in particular we will see that some hypersurfaces do not contract during this flow but straighten out and become totally geodesic i e in case n 1 s 1 they converge to a big s to be precise let g gij and a hi j be the induced metric and the second fundamental form on m and denote by h giih j a 12 h jh j the mean curvature and the squared norm of the second fundamental form respectively

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

flow by powers of the gauss curvature abstract we prove that convex hypersurfaces in r n 1 contracting under the flow by any power 1 n 2 of the gauss curvature converge after rescaling to fixed volume to a limit which is a smooth uniformly convex self similar contracting solution of the flow under additional central symmetry of the initial body we prove that the limit is the round sphere for 1

optimal vibration control of curved beams using distributed parameter models abstract the design of linear quadratic optimal controller using spectral factorization method is studied for vibration suppression of curved beam structures modeled as distributed parameter models the equations of motion for active control of the in plane vibration of a curved beam are developed firstly considering its shear deformation and rotary inertia and then the state space model of the curved beam is established directly using the partial differential equations of motion the functional gains for the distributed parameter model of curved beam are calculated by extending the spectral factorization method moreover the response of the closed loop control system is derived explicitly in frequency domain finally the suppression of the vibration at the free end of a cantilevered curved beam by point control moment is studied through numerical case studies in which the benefit of the presented method is shown by comparison with a constant gain velocity feedback control law and the performance of the presented method on avoidance of control spillover is demonstrated