on the soliton resolution for equivariant wave maps to the sphere we consider finite energy corotational wave maps with target manifold s2 we prove that for a sequence of times they decompose as a sum of decoupled harmonic maps in the light cone and a smooth wave map in the blowup case or a linear scattering term in the global case up to an error which tends to 0 in the energy space 2015 wiley periodicals inc

on the stability of the parallel sum of maximal monotone operators identifying operators with their graphs we study the continuity of the parallel sum of two monotone operators with respect to set convergence and we examine its stability from a quantitative point of view

rough solutions for the wave maps equation we consider the wave maps equation with values into a riemannian manifold which is isometrically embedded in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i our main result asserts that the cauchy problem is globally well posed for initial data which is small in the critical sobolev spaces this extends and completes recent work of tao and other authors

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

perturbation of nonlinear operators and their parallel sum on donne une version de la formule de kubo a b s lime 0 a ei b ei pour a et b des operateurs non lineaires

renormalization and blow up for the critical yang mills problem we consider the yangs mills equations in 4 1 dimensions this is the energy critical case and we show that it admits a family of solutions which blow up in finite time they are obtained by the spherically symmetric ansatz in the so 4 gauge group and result by rescaling of the instanton solution the rescaling is done via a prescribed rate which in this case is a modification of the self similar rate by a power of vertical bar log t vertical bar the powers themselves take any value exceeding 3 2 and thus form a continuum of distinct rates leading to blow up the methods are related to the authors previous work on wave maps and the energy critical semi linear equation however in contrast to these equations the linearized yang mills operator around an instanton exhibits a zero energy eigenvalue rather than a resonance this turns out to have far reaching consequences amongst which are a completely different family of rates leading to blow up logarithmic rather than polynomial corrections to the self similar rate c 2009 elsevier inc all rights reserved

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

the resolvent average for positive semidefinite matrices we define a new average termed the resolvent average for positive semidefinite matrices for positive definite matrices the resolvent average enjoys self duality and it interpolates between the harmonic and the arithmetic averages which it approaches when taking appropriate limits we compare the resolvent average to the geometric mean some applications to matrix functions are also given

2d segmentation using a robust active shape model with the em algorithm statistical shape models have been extensively used in a wide range of applications due to their effectiveness in providing prior shape information for object segmentation problems the most popular method is the active shape model asm however accurately fitting the shape model to an object boundary under a cluttered environment is a challenging task under such assumptions the model is often attracted toward invalid observations outliers leading to meaningless estimates of the object boundary in this paper we propose a novel algorithm that improves the robustness of asm in the presence of outliers the proposed framework assumes that both type of observations valid observations and outliers are detected in the image a new strategy is devised for treating the data in different ways depending on the observations being considered as valid or invalid the proposed algorithm assigns a different weight to each observation the shape parameters are recursively updated using the expectation maximization method allowing a correct and robust fit of the shape model to the object boundary in the image two estimation criteria are considered 1 the maximum likelihood criterion and 2 the maximum a posteriori criterion that use priors for the unknown parameters the methods are tested with synthetic and real images comprising medical images of the heart and image sequences of the lips the results are promising and show that this approach is robust in the presence of outliers leading to a significant improvement over the standard asm and other state of the art methods

isometries for the legendre fenchel transform it is shown that on the space of lower semicontinuous convex functions defined on r the conjugation map the legendre fenchel transform is an isometry with respect to some metrics consistent with the epi topology we also obtain isometries for the infinite dimensional case hilbert space and reflexive banach space but this time they correspond to topologies finer than the mosco epi topology

equivariant wave maps in two space dimensions singularities of corotational wave maps from 1 2 dimensional minkowski space into a surface n of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into in consequence for noncompact target surfaces of revolution the cauchy problem for wave maps is globally well posed 2003 wiley periodicals inc

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

global regularity of wave maps ii small energy in two dimensions we show that wave maps from minkowski space 1 n to a sphere sm 1 are globally smooth if the initial data is smooth and has small norm in the critical sobolev space in all dimensions n 5 this generalizes the results in the prequel 40 of this paper which addressed the high dimensional case n 5 in particular in two dimensions we have global regularity whenever the energy is small and global regularity for large data is thus reduced to demonstrating non concentration of energy

slow blow up solutions for the h 1 mathbb r 3 critical focusing semilinear wave equation given 1 2 and 0 arbitrary we prove the existence of energy solutions of ttu u u5 0 0 1 in r3 1 which blow up exactly at r t 0 as t 0 these solutions are radial and of the form u t 1 2w t r r t inside the cone r t where t t 1 w r 1 r2 3 1 2 is the stationary solution of 0 1 and is a radiation term with r t x t 2 t x t 2 x t 6 dx 0 t 0 outside of the light cone there is the energy bound r t u x t 2 ut x t 2 u x t 6 dx 0 the regularity of u increases with as in our accompanying article on wave maps 10 the argument is based on a renormalization method for the soliton profile w r

on the generalized parallel sum of two maximal monotone operators of gossez type d abstract the generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping in this article by assuming that the operators are maximal monotone of gossez type d we provide sufficient conditions of both interiority and closedness type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of gossez type d too this result will follow as a particular instance of a more general one concerning the maximal monotonicity of gossez type d of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

the parallel sum of nonlinear monotone operators on definit la somme parallele d operateurs non lineaires dans un espace de hilbert a l aide de leurs graphes

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc