solution of the cox thompson inverse scattering problem using finite set of phase shifts as ystem ofnonlinear equations is presented for the solution of the cox thompson inverse scattering problem 1970 j math phys 11 805 at fixed energy from a given finite set of phase shifts for physical angular momenta the nonlinear equations determine related sets of asymptotic normalization constants and nonphysical shifted angular momenta from which all quantities of interest including the inversion potential itself can be calculated as a first application of the method we use input data consisting of a finite set of phase shifts calculated from woods saxon and box potentials representing interactions with diffuse or sharp surfaces respectively the results for the inversion potentials their first moments and asymptotic properties are compared with those provided by the newton sabatier quantum inversion procedure it is found that in order to achieve inversion potentials of similar quality the cox thompson method requires a smaller set of phase shifts than th en ewton sabatier procedure

the plateau problem in alexandrov spaces we study the plateau problem of finding an area minimizing disk bounding a given jordan curve in alexandrov spaces with curvature these are complete metric spaces with a lower curvature bound given in terms of triangle comparison imposing an additional condition that is satisfied by all alexandrov spaces according to a conjecture of perel man we develop a harmonic map theory from two dimensional domains into these spaces in particular we show that the solution to the dirichlet problem from a disk is holder continuous in the interior and continuous up to the boundary using this theory we solve the plateau problem in this setting generalizing classical results in euclidean space due to j douglas and t rado and in riemannian manifolds due to c b morrey

hyperelliptic jacobians as billiard algebra of pencils of quadrics beyond poncelet porisms the thirty years old programme of griffiths and harris of understanding higher dimensional analogues of poncelet type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper starting with the observation of the billiard nature of some classical constructions and configurations we construct the billiard algebra that is a group structure on the set t of lines simultaneously tangent to d 1 quadrics from a given confocal family in the d dimensional euclidean space using this tool the related results of reid donagi and knorrer are further developed realized and simplified we derive a fundamental property of t any two lines from this set can be obtained from each other by at most d 1 billiard reflections at some quadrics from the confocal family we introduce two hierarchies of notions s skew lines in t and s weak poncelet trajectories s 1 0 d 2 the interrelations between billiard dynamics linear subspaces of intersections of quadrics and hyperelliptic jacobians developed in this paper enabled us to obtain higher dimensional and higher genera generalizations of several classical genus 1 results cayley s theorem weyr s theorem griffiths harris theorem and darboux theorem

tangent cones to positive 1 1 de rham currents we consider positive 1 1 de rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood without this assumption counterexamples to the uniqueness of tangent cones can be produced already in cn hence our result is optimal the key idea is an implementation for currents in an almost complex setting of the classical blow up of curves in algebraic or symplectic geometry unlike the classical approach in cn we cannot rely on plurisubharmonic potentials primary subject 53c38 secondary subjects 49q20 35j99

on nonsingular potentials of cox thompson inversion scheme we establish a condition for obtaining nonsingular potentials using the cox thompson inverse scattering method with one phase shift the anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying regge newton integral equation for the transformation kernel as a by product new inequality sequences of zeros of bessel functions are discovered

singular points of harmonic maps from 4 dimensional domains into 3 spheres in this article we show that if a stable stationary harmonic map u from a domain in r4 into s3 has an isolated singular point then the mapping degree deg u of u at is 1 or 1 furthermore the complete characterization of stable stationary tangent maps from b4 to s3 is given

nonuniqueness for the heat flow of harmonic maps on the disk we prove that for suitable initial data the heat flow of harmonic maps from the disk to the sphere admits infinitely many solutions characterised by backward bubbling at some arbitrarily large time all having uniformly bounded energy

erratum cayley type conditions for billiards within k quadrics in ℝd journal of physics a mathematical and general 2004 37 1269 1279 doi 10 1088 0305 4470 38 36 c01

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

blowup dynamics for smooth data equivariant solutions to the critical schrodinger map problem we consider the energy critical schrodinger map problem with the 2 sphere target for equivariant initial data of homotopy index k 1 we show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm which generates finite time blowup solutions we give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

singularity formation for the two dimensional harmonic map flow into s 2 we construct finite time blow up solutions to the 2 dimensional harmonic map flow into the sphere s 2 begin aligned u t delta u nabla u 2 u quad text in omega times 0 t u varphi quad text on partial omega times 0 t u cdot 0 u 0 quad text in omega end aligned where omega is a bounded smooth domain in mathbb r 2 u omega times 0 t rightarrow s 2 u 0 bar omega rightarrow s 2 is smooth and varphi u 0 big partial omega given any k points q 1 ldots q k in the domain we find initial and boundary data so that the solution blows up precisely at those points the profile around each point is close to an asymptotically singular scaling of a 1 corotational harmonic map we build a continuation after blow up as a h 1 weak solution with a finite number of discontinuities in space time by reverse bubbling which preserves the homotopy class of the solution after blow up furthermore we prove the codimension one stability of the one point blow up phenomenon

on nematic liquid crystals with variable degree of orientation energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals this mathematical model was recently posed by j l erickson in this paper we shall prove the existence local uniqueness and global regularity of minimizers we shall also estimate the size of possible defects

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

on the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2 sphere abstract let b 2 r 2 and s 2 r 3 be the unit disk and the unit sphere and let v b 2 r s 2 be a radially symmetric harmonic map heat flow whose singularities coincide with downward energy jumps then its finite time singularities are simple in the sense that precisely one harmonic sphere separates at a time

blow up and global existence for heat flows of harmonic maps in this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small

study of the cox thompson inverse scattering method with a coulomb potential in order to learn more about the precision of the inversion by the cox thompson method we investigated the inversion of phase shifts of a singular potential namely of a coulomb potential using asymptotically riccati bessel functions as reference functions we could only approximately reproduce the singularity of the coulomb potential at the origin we also show uncertainties in the inverted potential due to different minima in the minimization solution of the nonlinear equations of the cox thompson procedure as a result we conclude that one has to take much care with the inversion of experimental phase shifts suffering from measurement errors

formal asymptotics of bubbling in the harmonic map heat flow the harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry we present an analysis of the asymptotic behavior of singularities arising in this flow for a special class of solutions which generalizes a known radially symmetric reduction specifically the rate at which blowup occurs is investigated in settings with certain symmetries using the method of matched asymptotic expansions we identify a range of blowup scenarios in both finite and infinite time including degenerate cases

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

partial regularity for weak heat flows into spheres in this paper we show that a weak heat flow of harmonic maps from a compact riemannian manifold possibly with boundary into a sphere satisfying the monotonicity inequality and the energy inequality is regular off a closed set of m dimensional hausdorff measure zero

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

singularities of energy minimizing maps from the ball to the sphere examples counterexamples and bounds we study maps qp from domains q in r3 to s2 which have prescribed boundary value functions 4 and which minimize dirichlet s integral energy such p s can have isolated singular points and our principal concern is the number and arrangement of such singularities our main result is that the number of singular points is bounded above by a constant times dirichlet s integral of 4 on d q and we show that this linear law is the best possible in two different senses furthermore singularities of minimizing q s sometimes have peculiar and counterintuitive properties i we illustrate by example that the mapping area jacobian integral of 4 can be zero although qp has many singular points ii we also show how to construct uniquely minimizing qp s having many singular points stacked up vertically near d q like bubbles in a pan of water about to boil iii finally we show by example that symmetries of q and 4 do not insure corresponding symmetries of qp and its singularities in particular singularities in p can be unstable under small perturbations of 4

mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere we describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere then we use this as a test case to compute optimal meshes in presence of sharp boundary layers for the well posedness of the continuous minimizing problem we introduce a lower semicontinuous extension of the energy with respect to weak convergence in bv and we prove that the extended minimization problem has a unique singular solution we then show how a moving finite element method in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization mimics this geometric point of view finally we present numerical computations with boundary layers of zero thickness and we give numerical evidence of the convergence of the method this last aspect is proved in another paper

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

axially symmetric harmonic maps minimizing a relaxed energy here we obtain various results on the class of axially symmetric harmonic maps from b3 to s2 we find some new classes of non minimizing harmonic maps exhibiting unusual singular behavior optimal partial regularity estimates are obtained for mappings which minimize among axially symmetric maps various relaxed energies which have been studied in 4 and 11

global existence and blow up for harmonic map heat flow abstract for degree one equivariant maps on bounded domains the question of finite time blow up vs global existence of solutions to the harmonic map heat flow has been well studied in this paper we study the cauchy problem for degree m equivariant harmonic map heat flow from 2 1 dimensional space time into the 2 sphere with initial energy close to the energy of harmonic maps it is proved that solutions are globally smooth for m 4 whereas for m 1 we show that finite time singularities can form for this class of data

blow up dynamics for equivariant critical schrodinger maps for the schrodinger map equation u t u times triangle u rm in mathbb r 2 1 with values in s2 we prove for any nu 1 the existence of equivariant finite time blow up solutions of the form u x t phi lambda t x zeta x t where phi is a lowest energy steady state lambda t t 1 2 nu and zeta t is arbitrary small in dot h 1 cap dot h 2

on the radial behavior of minimal surfaces and the uniqueness of their tangent cones suppose v is an m 1 dimensional minimal surface in r containing 0 as an isolated singular point what can one say about the structure of v near o this question initiated the present investigation and in certain special cases has a satisfactory answer 1 if one of the tangent cones to v at 0 is of the form 0 m where m is an m dimensional minimal submanifold of s and if also for each jacobi normal vectorfield z of m in sn 1 there is a one parameter family of minimal surfaces in s 1 having velocity z at m then 0 m is the unique tangent cone to v at 0 with v converging to 0 m for small radii r with rate rx it 0 2 in case m is the cartesian product of two standard spheres of appropriate radii then each jacobi vectorfield on m arises from isometric motions of sn 1 and the jacobi vectorfield hypothesis in 1 is satisfied this is shown in chapter 6 the uniqueness of tangent cones to one dimensional stationary varifolds with positive densities was shown in aa and the uniqueness of tangent cones to certain two dimensional area minimizing singular surfaces in r3 was shown by j taylor in tj1i tj2 tj3 also b white has shown such uniqueness for area minimizing hypersurfaces mod 4 in space dimen

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

type ii blow up for the four dimensional energy critical semi linear heat equation abstract we consider the energy critical four dimensional semi linear heat equation t u u u 3 0 we show the existence of type ii finite time blow up solutions and give a sharp description of the corresponding singularity formation these solutions concentrate a universal bubble of energy in the critical topology u t r 1 q r t u in h 1 where the blow up profile is given by the talenti aubin soliton q r 1 1 r 2 8 and with speed t t t log t t 2 as t t our approach uses a robust energy method approach developed for the study of geometrical dispersive problems raphael and rodnianski 2012 18 merle et al 2011 15 and lies in the continuation of the study of the energy critical harmonic heat flow raphael and schweyer 2011 19 and the energy critical four dimensional wave equation hillaret and raphael 2010 5

the inverse scattering problem at fixed energy based on the marchenko equation for an auxiliary sturm liouville operator a new approach is proposed to the solution of the quantum mechanical inverse scattering problem at a fixed energy the method relates the fixed energy phase shifts to those arising in an auxiliary sturm liouville problem via the interpolation theory of the weyl titchmarsh m function then a marchenko equation is solved to obtain the potential as an illustration test results are shown including the reconstructions of the piecewise constant and coulomb type potentials a byproduct of the new framework is a demonstration of the connection found between the gelfand levitan and marchenko formalisms

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we