analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

cartesian boundary treatment of curved walls for high order computational aeroacoustics schemes it is known that the use of high order central difference schemes on a cartesian grid is preferable for the computation of acoustic wave propagation problems those schemes tend to be less dispersive and dissipative than most other types of schemes they are also more capable of providing an accurate wave speed a cartesian boundary treatment for problems involving the scattering of acoustic waves by solid objects with curved boundary surfaces designed to be used in conjunction with such high order central difference schemes is proposed the development of this method is based on the observation that a solid wall actually exerts a pressure force on the fluid to keep it from flowing across the wall surface in this method ghost values of pressure are introduced at mesh points adjacent to the solid boundary inside the object the ghost values are then chosen so that the solid wall boundary condition is satisfied the method is also applicable to objects with sharp corners numerical examples are provided

biharmonic maps from r 4 into a riemannian manifold for a domain r 4 and a compact riemannian manifold n r k without boundary if u w 2 2 n is an extrinsic or intrinsic respectively biharmonic map then u c n

a note on the radiation boundary conditions for the helmholtz equation the radiation boundary conditions in two dimensions are derived in the body fitted coordinate system using the method of successive approximations for the wave envelope function results are valid for a convex scatterer with continuous radius of curvature on the surface of the scatterer in the special case when the computational boundary is circular the boundary operators derived are identical to the bayliss operators the on surface radiation condition is examined it is shown that for a large conducting circular cylinder a boundary operator of infinite order should be used due to the breakdown of the asymptotic expansions of the boundary operators on the surface of the scatterer the leading order result based on the boundary operator of infinite order applied on the surface of the cylinder is the same as the result obtained by the method of physical optics

a relaxation of the intrinsic biharmonic energy the tension field u of a map u from a domain omega subset mathbb r m into a manifold n is the negative l2 gradient of the dirichlet energy in this paper we study the intrinsic biharmonic energy functional t u int omega tau u 2 in order to overcome the lack of coercivity of t we extend it to a larger space we construct minimizers of the extended functional via the direct method and we study the relation between these minimizers and critical points of t our results are restricted to dimensions m 4

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

quasiregular mappings in even dimensions o introduction 1 notation 2 some exterior algebra 3 differential forms in lnm i2 a n 4 differential systems for quasiregular mappings 5 liouville theorem in even dimensions 6 hodge theory in lp r n 7 the beltrami equation in even dimensions 8 the beurling ahlfors operator 9 regularity theorems for quasiregular mappings 10 the caccioppoli type estimate 11 removability theorems for quasiregular mappings 12 some examples appendix the 4 dimensional case

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

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

regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps let omega subset mathbb r n n ge 2 be open and n subset mathbb r k a smooth compact riemannian manifold without boundary we consider the approximated harmonic map equation delta u a u nabla u nabla u f for maps u in h 1 omega n where f in l p omega mathbb r k for p frac n 2 we prove holder continuity for weak solution s which satisfy a certain smallness condition for p frac n 2 we derive an energy estimate which allows to prove partial regularity for stationary solutions of the heat flow for harmonic maps in dimension n le 4

stratification for the singular set of approximate harmonic maps the aim of this note is to extend the results in naber and valtorta ann math 2 185 131 227 https doi org 10 4007 annals 2017 185 1 3 2017 to the case of approximate harmonic maps more precisely we will proved that the singular strata mathcal s k u of an approximate harmonic map are k rectifiable and we will show effect bounds on the quantitative strata in the process we will simplify many of the arguments from naber and valtorta 2017 and in particular we produce a new main covering lemmas which vastly simplifies the older argument

nonstationary weak limit of a stationary harmonic map sequence let m and n be two compact riemannian manifolds let uk be a sequence of stationary harmonic maps from m to n with bounded energies we may assume that it converges weakly to a weakly harmonic map u in h1 2 m n as k in this paper we construct an example to show that the limit map u may not be stationary 2002 wiley periodicals inc

energy identity of approximate biharmonic maps to riemannian manifolds and its application abstract we consider in dimension four weakly convergent sequences of approximate biharmonic maps to a riemannian manifold with bi tension fields bounded in l p for p 4 3 we prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on r 4 as a corollary we obtain an energy identity for the heat flow of biharmonic maps at time infinity

stationary biharmonic maps from rm into a riemannian manifold for m 5 we prove that a stationary extrinsic or intrinsic respectively biharmonic map u w2 2 n from rm into a riemnanian manifold n is smooth away from a closed set of m 4 dimensional hausdorff measure zero 2003 wiley periodicals inc

a regularity theory of biharmonic maps in this article we prove the regularity of weakly biharmonic maps of domains in euclidean four space into spheres as well as the corresponding partial regu larity result of stationary biharmonic maps of higher dimensional domains into spheres c 1999 john wiley sons inc in this article we consider for simplicity the class of bihar monic maps from eu clidean domains to spheres we realize the standard spheres s k as unit vectors in r k 1 and consider maps u s k as vector valued functions that are contained in s k the energy functional for biharmonic maps is then j uj 2 dx a locally de fined biharmonic map is a map that is critical with respect to c ompactly supported variations we note that in the case where the domain has dimension four this en ergy functional is conformally invariant and hence conformal maps of euclidean four space are biharmonic in this sense we remark that this definition of bihar monic map depends on the embedding of the target space in euclidean space we

biharmonic submanifolds of mathbb c p n we give some general results on proper biharmonic submanifolds of a complex space form and in particular of the complex projective space these results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field we find the relation between the bitension field of the inclusion of a submanifold bar m in mathbb c p n and the bitension field of the inclusion of the corresponding hopf tube in mathbb s 2n 1 using this relation we produce new families of proper biharmonic submanifolds of mathbb c p n we study the geometry of biharmonic curves of mathbb c p n and we characterize the proper biharmonic curves in terms of their curvatures and complex torsions

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

nonlinear partial differential equations of fourth order under mixed boundary conditions solutions to nonlinear partial differential equations of fourth order are studied boundary regularity is proved for solutions that satisfy mixed boundary conditions various geometric situations including so called triple points are considered regularity is measured in sobolev slobodeckii spaces and the results are sharp in this scale the approach is based on the use of a first order difference quotient method

application of the integral equation asymptotic phase method to two dimensional scattering a hybrid procedure called the integral equation asymptotic phase ie ap method is investigated for scattering from perfectly conducting cylinders of arbitrary cross section shape the ie ap approach employs an asymptotic solution to predict the relatively rapid phase dependence of the unknown current distribution to leave a slowly varying residual function that can be represented by a coarse density of unknowns in the present investigation the current density appearing within the combined field integral equation is replaced by the product of a rapidly varying phase function obtained from the physical optics current and a residual function the resulting equation is discretized by the method of moments using subsectional quadratic polynomial basis functions defined on curved cells to represent the residual function results show that the required density of unknowns can often be as few as one per wavelength on average without a significant loss of accuracy in the computed current density even for scatterers with corners

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

partial regularity for stationary harmonic maps at a free boundary let and be smooth riemannian manifolds of the dimension n 2 with nonempty boundary and compact without boundary we consider stationary harmonic maps u h 1 with a free boundary condition of the type u given a submanifold we prove partial boundary regularity namely sing u 0 a result that was until now only known in the interior of the domain see b the key of the proof is a new lemma that allows an extension of u by a reflection construction once the partial regularity theorem is known it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

remarks on the regularity of biharmonic maps in four dimensions we study intrinsic biharmonic maps on a four dimensional domain into a smooth compact riemannian manifold we prove a partial regularity result without the assumption that the second derivatives are square integrable

the regularity of a semilinear elliptic system with quadratic growth of gradient abstract in this paper we study semilinear elliptic systems with critical nonlinearity of the form 0 1 u q x u u for u r n r k q has quadratic growth in u our work is motivated by elliptic systems for harmonic map and biharmonic map when n 2 such a system does not have smooth regularity in general for w 1 2 weak solutions by a well known example of j frehse classical results of harmonic map proved by f helein for n 2 and f bethuel for n 3 assert that a w 1 n weak solution of harmonic map is always smooth we extend bethuel s result to general system 0 1 that a w 1 n weak solution of the system is smooth for n 3 for a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map we prove a similar result that a w 2 n 2 weak solution of such system is always smooth for n 5 we also construct various examples and these examples show that our regularity results are optimal in various sense

regularity of n 2 harmonic maps into spheres we prove holder continuity for n 2 harmonic maps from arbitrary subsets of rn into a sphere this extends a recent one dimensional result by f da lio and t riviere to arbitrary dimensions the proof relies on compensation effects which we quantify adapting an approach for wente s inequality by l tartar instead of besov space arguments which were used in the one dimensional case moreover fractional analogues of hodge decomposition and higher order poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied this work was the author s phd thesis written in march 2010

energy concentration for almost harmonic maps and the willmore functional let be an open domain in 3 or 4 and n a smooth compact riemannian manifold we consider the dirichlet energy e u for maps u n and its negative l2 gradient the tension field u we study sequences of maps ui n with open image in new window if the maps are sufficiently regular we find strong h1 subconvergence away from a generalized submanifold in if the limit map is regular too we can estimate a willmore type energy of this generalized submanifold

numerical solution for the scattering of sound waves by a circular cylinder on introduit une amplitude de diffusion generalisee et on l utilise pour calculer une equation de helmholtz transformee couplee avec la condition de rayonnement et les conditions aux limites a la surface du cylindre l equation de helmholtz transformee est resolue par une methode de differences finies

neck analysis for biharmonic maps in this paper we study the blow up of a sequence of both extrinsic and intrinsic biharmonic maps in dimension four with bounded energy and show that there is no neck in this process moreover we apply the method to provide new proofs to the removable singularity theorem and energy identity theorem of biharmonic maps

blow up analysis and boundary regularity for variationally biharmonic maps abstract we consider critical points u n of the bi energy u 2 d x where r m is a bounded smooth domain of dimension m 5 and n r l a compact submanifold without boundary more precisely we consider variationally biharmonic maps u w 2 2 n which are defined as critical points of the bi energy that satisfy a certain stationarity condition up to the boundary for weakly convergent sequences of variationally biharmonic maps we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non constant biharmonic 4 spheres or 4 halfspheres in the target manifold as an application we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist

time dependent wave envelope finite difference analysis of sound propagation a transient finite difference wave envelope formulation is presented for sound propagation without steady flow before the finite difference equations are formulated the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction this transformation reduces the required number of grid points by an order of magnitude physically the transformed pressure represents the amplitude of the conventional sound wave the derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements to illustrate the method example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct the numerical results are in good agreement with exact analytical results

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

a monotonicity formula for stationary biharmonic maps we give a rigorous proof of the monotonicity formula of s y a chang l wang and p yang 3 for extrinsically stationary biharmonic maps of class w2 2

reeb dynamics inspired by katok s example in finsler geometry inspired by katok s examples of finsler metrics with a small number of closed geodesics we present two results on reeb flows with finitely many periodic orbits the first result is concerned with a contact geometric description of magnetic flows on the 2 sphere found recently by benedetti we give a simple interpretation of that work in terms of a quaternionic symmetry in the second part we use hamiltonian circle actions on symplectic manifolds to produce compact connected contact manifolds in dimension at least five with arbitrarily large numbers of periodic reeb orbits this contrasts sharply with recent work by cristofaro gardiner hutchings and pomerleano on reeb flows in dimension three with the help of hamiltonian plugs and a surgery construction due to laudenbach we reprove a result of cieliebak one can produce hamiltonian flows in dimension at least five with any number of periodic orbits in dimension three with any number greater than one

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

a second order variational problem with a lack of coercivity for n and 2n n 2 p 2n 4 n 4 we study the functional e2 u 12 u u p 2u 2dx in an appropriate nonlinear space the minimization problem can be solved using the direct method we also construct solutions of the gradient flow

bubble tree convergence for harmonic maps let be a compact riemann surface any sequence fn m of harmonic maps with bounded energy has a bubble tree limit consisting of a harmonic map o m and a tree of bubbles fk s2 m we give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class and that the images of the fn converge pointwise we then give explicit counterexamples showing that bubble tree convergence fails i for harmonic maps fn when the conformal structure of varies with n and ii when the conformal structure is fixed and n is a palais smale sequence for the harmonic map energy

contact spheres and hyperkahler geometry a taut contact sphere on a 3 manifold is a linear 2 sphere of contact forms all defining the same volume form in the present paper we completely determine the moduli of taut contact spheres on compact left quotients of su 2 the only closed manifolds admitting such structures we also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles

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