removability of singular sets of harmonic maps it is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density moreover the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability this work generalizes the results in cl df lg12 which were proved under the assumption that the singular sets are isolated points or smooth submanifolds

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

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

an induction principle for the weighted p energy minimality of x x abstract in this paper we investigate minimizing properties of the map x x from the euclidean unit ball b n to its boundary s n 1 for the weighted energy functionals e p n u b n x u p d x where p 2 we establish the following induction principle if the map x x b n 1 s n minimizes e p n 1 among maps u b n 1 s n satisfying u x x on s n then the map y y b n s n 1 minimizes e p 1 n among maps v b n s n 1 satisfying v y y on s n 1 this result enables us to enlarge the range of values of p and for which x x minimizes e p n

on the solvability of asymptotically linear systems at resonance abstract this paper is concerned with the solvability of the system u 1 1 v f x u v h 1 x in v 1 2 u g x u v h 2 x in u v 0 on at resonance at the simple eigenvalue 1 of the corresponding linear eigenvalue problem here r n n 1 is a bounded domain with c 2 boundary 0 1 a bounded interval if n 1 and 1 2 are positive constants the nonlinear perturbations f x u v g x u v r 2 r are caratheodory functions that are sublinear at infinity we employ the lyapunov schmidt method to provide sufficient conditions on h 1 h 2 l r r n to guarantee the solvability of the system

stability and singularities of harmonic maps into 3 spheres the behavior of harmonic maps from a domain in r4 into s3 is discussed in this paper we show that if u is a strictly stable stationary harmonic map b4 s3 such that the singular set sing u of u consists of 0 then the degree deg u 0 of u at 0 is zero and that if u is a weakly stable harmonic map b4 s3 with sing u 0 then deg u 0 0 or 1 furthermore the characterization of homogeneous stable stationary harmonic maps from b4 into s3 is given

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

solving variational problems and partial differential equations mapping into general target manifolds a framework for solving variational problems and partial differential equations that define maps onto a given generic manifold is introduced in this paper we discuss the framework for arbitrary target manifolds while the domain manifold problem was addressed in j comput phys 174 2 2001 759 the key idea is to implicitly represent the target manifold as the level set of a higher dimensional function and then implement the equations in the cartesian coordinate system where this embedding function is defined in the case of variational problems we restrict the search of the minimizing map to the class of maps whose target is the level set of interest in the case of partial differential equations we re write all the equation s geometric characteristics with respect to the embedding function we then obtain a set of equations that while defined on the whole euclidean space are intrinsic to the implicitly defined target manifold and map into it this permits the use of classical numerical techniques in cartesian grids regardless of the geometry of the target manifold the extension to open surfaces and submanifolds is addressed in this paper as well in the latter case the submanifold is defined as the intersection of two higher dimensional hypersurfaces and all the computations are restricted to this intersection examples of the applications of the framework here described include harmonic maps in liquid crystals where the target manifold is a hypersphere probability maps where the target manifold is a hyperplane chroma enhancement texture mapping and general geometric mapping between high dimensional manifolds

the existence results for optimal control problems governed by a variational inequality in this paper we establish the existence of the optimal control for an optimal control problem where the state of the system is defined by a variational inequality problem with monotone type mappings moreover as an application we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle

a global bifurcation result for a class of semipositone elliptic systems abstract we consider a system of the form u 1 v f v in v 2 u g u in u 0 v on where s def max s 0 1 and 2 are fixed positive constants r is the bifurcation parameter and r n n 1 is a bounded domain with smooth boundary a bounded open interval if n 1 the nonlinearities f g r r are continuous functions that are bounded from below sublinear at infinity and have semipositone structure at the origin f 0 g 0 0 we show that there are two disjoint unbounded connected components of the solution set and discuss the nodal properties of solutions on these components finally as a consequence of these results we infer the existence and multiplicity of solutions for in a neighborhood containing the simple eigenvalue of the associated eigenvalue problem

existence of solutions for quasilinear elliptic obstacle problems two existence results for a class of obstacle problems are obtained by using a result developed in this paper

existence of solutions for a class of elliptic variational inequalities in this paper the existence of solutions to the variational inequalities involving the p laplacian type operator div j u on an unbounded domain in rn is discussed

resonance problems for the p laplacian abstract we consider resonance problems at an arbitrary eigenvalue of the p laplacian and prove the existence of weak solutions assuming a standard landesman lazer condition we use variational arguments to characterize certain eigenvalues and then to establish the solvability of the given boundary value problem

smooth dependence on parameters of solutions and contact regions for an obstacle problem we consider a class of variational inequalities which includes a model of a beam compressed by a force and unilaterally supported by a connected obstacle at the height h a smooth dependence on parameters and h of solutions and intervals of their contact with the obstacle is proved in a neighbourhood of a given solution the basic idea is to show local equivalence of the inequality with a suitable equation and to apply the implicit function theorem to this equation

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

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