Harmonic map heat flow for axially symmetric data

We examine the harmonic map heat flow problem for maps between the three-dimensional ball and the two-sphere. We give blow-up results for certain initial data. We establish convergence results for suitable axially symmetric initial data, and discuss generalizations to higher dimensions.     

Axially symmetric harmonic maps minimizing a relaxed energy

Here we obtain various results on the class of axially symmetric harmonic maps from B ³ to S ². 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].     

Exponentially harmonic maps and their properties

We show that the associated quadratic differentials of exponentially harmonic maps are holomorphic under certain circumstance. We study the sufficient and necessary conditions for axially symmetric maps which are exponentially harmonic. We investigate exponentially harmonic equations for rotationally symmetric maps between rotationally symmetric manifolds of low dimensions.     

On rotating black holes in equilibrium in general relativity

We study asymptotically flat axially symmetric stationary solutions of the Einstein vacuum equations. These represent rotating black holes in equilibrium. The equations reduce outside the axis of symmetry to a harmonic map problem into the hyperbolic plane, with prescribed rates of blow-up for the map on the axis and at infinity as boundary conditions. We prove existence and uniqueness of solutions in the case of zero total angular momentum.     

The maximum principle and the Dirichlet problem for Dirac-harmonic maps

We establish a maximum principle and uniqueness for Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold N. Then we prove an existence theorem for a boundary value problem for Dirac-harmonic maps.     

Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.     

Existence and Liouville theorems for V -harmonic maps from complete manifolds

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.     

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

The Stefan problem for the Fisher–KPP equation

We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.     

The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

     

Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations

In this paper, we consider the harmonic maps with potential from compact Riemannian manifold with boundary into a convex ball in any Riemannian manifold. We will establish some general properties such as the maximum principles, uniqueness and existence for these maps, and as an application of them, we derive existence and uniqueness result for the Dirichlet problem of the Landau-Lifshitz equations. Received: December 10, 1997 / Accepted: June 29, 1998     

Théorèmes d'existence pour un modèle membranaire « proprement invariantde plaque non linéairement élastique

We establish the “local” existence of an injective solution to the nonlinear, “properly invariant”, membrane plate model, stated in [1] and [2], successively for the clamped plate submitted to forces parallel to its plane and for the plate submitted to a boundary condition of place of “extended” state.     

A Global Existence Result For The Heat Flow of Harmonic Maps

We consider the heat flow for harmonic maps from R^m to a compact manifold N. When the L^m norm of the gradient of the initial data is small, we prove the existence of a global solution. We prove a similar result for the boundary value problem, when the boundary of the manifold M maps into a point.     

Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space to a rank one symmetric space N of noncompact type with singular boundary data . Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously. Received: March 11, 1999 / Accepted April 23, 1999     

Heat Transfer with a Free Boundary Moving Within a Concentrated Thermal Capacity

A finite-difference scheme is described for the numerical solutionof plane-parallel and axially symmetric problems of heat transferwith a free boundary moving within concentrated thermal capacities.The relevant existence and uniqueness theorems are proved inFasano, Primicerio & Rubinstein (1980) and Rubinstein (1980).Numerical illustrations are offered.     

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ?(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

Existence for VT-harmonic maps from compact manifolds with boundary

We consider a kind of generalized harmonic maps, namely, the VT-harmonic maps. We prove an existence theorem for the Dirichlet problem of VT-harmonic maps from compact manifolds with boundary.

     

N-Black hole stationary and axially symmetric solutions of the einstein/maxwell equations

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities φ?³\Σ→² _c where σ os a subset of the axis of symmetry, and ² _c is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities φ?ⁿ\σ→ where Σ is a submanifold of ?ⁿ of co-dimension ≥ 2,and  is a classical Riemannian globally symmetric space of noncompact typeand rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular starts.     

Linearization of a free boundary problem in corrosion detection

We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part ΓI⊂∂Ω of the boundary of a bounded, planar domain Ω from one Cauchy data pair (u,∂u/∂ν) of a harmonic potential u in Ω collected on an accessible boundary subset ΓA⊂∂Ω. We prove Fréchet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem.