on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

elliptic equations strongly degenerate at a point abstract in this paper we will mainly propose some problems for a class of degenerate elliptic equations either linear or nonlinear we will study some special cases of these problems and reveal some phenomena which may not have been noticed previously our problems originated from the self similar solutions of the heat flow of harmonic maps we will prove that the self similar solutions or the so called quasi harmonic spheres are discontinuous at infinity for the equivariant case in other words the equivariant quasi harmonic spheres are not continuous images of topological spheres

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

on stable complete minimal hypersurfaces in r n 1 abstract by using the interior curvature estimate and gromov s compactness theorem we prove that a stable complete minimal hypersurface in rn l with finite total scalar curvature must be a hyperplane this extends the result obtained by p b rard in remarques sur l quation de j simons pitman monographs surveys pure appl math vol 52 1991 pp 47 57

the scaled flexibility matrix method for the efficient solution of boundary value problems in 2d and 3d layered elastic media we present a method that extends the flexibility matrix method for multilayer elasticity problems to include problems with very thin layers this method is particularly important for solving problems in which one or a number of very thin layers are juxtaposed with very thick layers the standard flexibility matrix method suffers from round off errors and poor scaling of the flexibility equations which occur when one of the layers in the multilayered medium becomes much smaller than the others the method proposed in this paper makes use of power series expansions of the various components of the flexibility matrix in order to arrive at a system of equations that is appropriately scaled the effectiveness of the scaled flexibility matrix method is demonstrated on a number of test problems

homotopy classes of harmonic maps of the stratified 2 spheres and applications to geometric flows abstract we show that the set of harmonic maps from the 2 dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes we apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2 sphere in the connected sum of two closed 3 dimensional manifolds m 1 s 3 and m 2 s 3 r p 3 which must develop finite time singularity

some bernstein type rigidity theorems abstract we prove some bernstein type rigidity theorems for complete submanifolds in a euclidean space and space like submanifolds of a lorentzian space in particular we obtain a bernstein rigidity theorem for complete minimal submanifolds of arbitrary codimension in euclidean space

gap theorems for minimal submanifolds of a hyperbolic space abstract this paper provides some gap theorems for complete immersed minimal submanifolds of dimension no less than five in a hyperbolic space namely we show that an n 5 dimensional complete immersed minimal submanifold m in a hyperbolic space is totally geodesic if the l 2 norm of a on geodesic balls centered at some point p m has less than quadratic growth and if either sup x m a 2 x is not too large or the l n norm of a on m is small here a is the second fundamental form of m

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

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

brownian motion and the formation of singularities in the heat flow for harmonic maps we develop a general framework for a stochastic interpretation of certain nonlinear pdes on manifolds the linear operation of takin expectations is replaced by the concept of martingale means namely the notion of deterministic starting points of martingales with respect to the levi civita connection ending up at a prescribed state we formulate a monotonicity condition for the riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation such estimates lead to simple criteria for blow ups in the nonlinear heat flow for harmonic maps with small initial energy

finite time singularity of the nematic liquid crystal flow in dimension three in this paper we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity the first example is constructed within the class of axisymmetric solutions while the second example is constructed for any generic initial data u 0 d 0 that has sufficiently small energy and d 0 has a nontrivial topology

the heat flow of harmonic maps from noncompact manifolds the authors consider the global existence of the heat flow of harmonic maps from noncompact manifolds while imposing restrictions on the initial data

global existence of the equivariant yang mills heat flow in four space dimensions we show that the yang mills heat flow of so 4 equivariant connections on an su 2 bundle over a ball in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i admits a smooth solution for all time for any initial and boundary data of finite energy

a dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems we present a novel multigrid mg procedure for the efficient solution of the large non symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi layered elastic medium the governing equations involve a highly non linear coupled system of integro partial differential equations along with the fracture front free boundary problem the conditioning of the algebraic equations typically degrades as o n3 a number of characteristics of this problem present significant new challenges for designing an effective mg strategy large changes in the coefficients of the pde are dealt with by taking the appropriate harmonic averages of the discrete coefficients coarse level green s functions for multiple elastic layers are constructed using a single dual mesh and superposition coarse grids that are sub sets of the finest grid are used to treat mixed variable problems associated with pinch points localized approximations to the jacobian at each mg level are used to devise efficient gauss seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements the performance of the mg preconditioner is demonstrated in a number of numerical experiments copyright 2005 john wiley sons ltd

well posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data we investigate the well posedness of 1 the heat flow of harmonic maps from mathbb r n to a compact riemannian manifold n without boundary for initial data in bmo and 2 the hydrodynamic flow u d of nematic liquid crystals on mathbb r n for initial data in bmo 1 bmo

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

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

an implicit algorithm for the propagation of a hydraulic fracture with a fluid lag abstract we present an implicit moving mesh algorithm for the study of the propagation of plane strain hydraulic fracture in an impermeable medium in particular the fluid front is allowed to lag behind the fracture tip the solution expressed in the proper scaling evolves for a given value of a dimensionless toughness from a zero stress zero time self similar solution characterized by a finite lag to a large stress large time self similar solution with zero lag a numerical solution for the transient problem is presented using two distinct meshes stretching at different velocities for the lag and fluid filled part of the crack the lubrication and elasticity equations are solved in a coupled manner using an implicit scheme the lag size also coupled to the preceding equations is obtained via the fracture propagation condition the numerical results are discussed and compared with the known zero lag solution for large stress large time