Local Uniqueness Theorem Research Articles

Holonomy and projective symmetry in spacetimes

It is shown using a spacetime curvature classification and decomposition that for certain holonomy types of a spacetime, proper projective vector fields cannot exist. Existence is confirmed, by example, for the remaining holonomy types. In all except the most general holonomy type, a local uniqueness theorem for proper projective symmetry is established.

Local Existence for the Dumbbell Model of Polymeric Fluids

A local existence and uniqueness theorem is proved for a micro-macro model for polymeric fluid, as well as the property of the solution. The polymer stress tensor is given by an integral which involves the solution of a diffusion equation, the coefficient of this diffusion equation depends on the gradient of the solution of the Navier–Stokes equation.

On the Boussinesq Flow with Nondecaying Initial Data

This paper is concerned with the Boussinesq equations which describe the heat convection in a viscous incompressible fluid. Local existence and uniqueness theorems are established for the n-dimensional Boussinesq equations in the whole space with nondecaying initial data. In two dimensional case the solution can be extended globally in time without smallness of the initial data.

Extremal isolated horizons: a local uniqueness theorem

We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr–Newman solution on the event horizon. We also discuss the general case of a symmetric, extremal isolated horizon. In particular, we analyse the case of a two-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal ones, is explained.

ON DETERMINATION OF A GAUGE FIELD ON ℝd FROM ITS NON-ABELIAN RADON TRANSFORM ALONG ORIENTED STRAIGHT LINES

We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radon transform along oriented straight lines. The determination is considered modulo gauge transformations. Our results include: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs (i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction to the attenuated X-ray transform.AMS 2000 Mathematics subject classification: Primary 53C65; 81U40

Viscosity Solutions of the Bellman Equation for Exit Time Optimal Control Problems with Vanishing Lagrangians

We study the Hamilton--Jacobi--Bellman equation for undiscounted exit time optimal control problems for fully nonlinear systems and fully nonlinear singular Lagrangians using the dynamic programming approach. We prove a local uniqueness theorem characterizing the value functions for these problems as the unique viscosity solutions of the corresponding Hamilton--Jacobi--Bellman equations that satisfy appropriate boundary conditions. The novelty of this theorem is in the relaxed hypotheses on the lower bound on the Lagrangian and the very general assumptions on the target set. As a corollary, we show that the value function for the Fuller problem is the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation that vanishes at the origin and satisfies certain growth conditions. This implies as special cases first that the value function of this problem is the unique proper viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation, in the class of all functions which are continuous in the plane and null at the origin, and second that this value function is the unique viscosity solution of that equation in a class which includes functions which are not bounded below. We also apply our results to the degenerate eikonal equation of geometric optics and to the shape-from-shading equations in image processing. Our theorem also applies to problems with noncompact targets and unbounded control sets whose Lagrangians take negative values.

Global existence of solutions to fuzzy differential equations

The local existence and uniqueness theorems are investigated by Wu et al. (J. Math. Anal. Appl. 202 (1996) 629–644), for the Cauchy problem of the fuzzy-valued mappings of a real variable whose values are in the fuzzy number space ( E n,D ). In this paper, we point out a variety of results which assure global existence of solutions to fuzzy differential equations.

Incompressible Flows of an Ideal Fluid¶with Unbounded Vorticity

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in Rn singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness.

On the inverse problem of determining a connection on a vector bundle

Abstract - We consider the problem of determining a connection on a vector bundle over a compact Riemannian manifold with boundary from the known parallel transport between boundary points along geodesics. The main result is the local uniqueness theorem: if two connections ∇' and ∇'' are C-close to a given connection ∇ whose curvature tensor is sufficiently small, then the coincidence of parallel transports with respect to ∇' and ∇'' implies the existence of an automorphism of the bundle which is identical on the boundary and transforms ∇' to ∇'' . A linearized version of the problem is also considered.

An analysis of the scalar geodetic boundary-value problem with natural regularity results

A new local existence and uniqueness theorem is obtained for the scalar geodetic boundary-value problem in spherical coordinates. The regularities H-alpha and H1+alpha are assumed for the boundary data g (gravity) and v (gravitational potential) respectively.

Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations

We prove a local existence and uniqueness theorem for abstract parabolic problems of the type x = Ax-{-f(t, x) when the nonlinearity f satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations. ©1999 American Mathematical Society.

Strong Uniqueness for Schrödinger Operators with Kato Potentials

We prove a strong unique continuation result for Schrödinger inequalities, i.e., we obtain that a flat u so that |Δu| ≤|Vu| should be zero, provided that V is a radial Kato potential. It gives an extension of a result by E. B. Fabes, N. Garofalo and F. H. Lin [3] who got a weak local uniqueness theorem. Our method relies on sharp Carleman estimates.

Successive approximations of solutions to stochastic functional differential equations

In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.

Nonlinear stochastic functional integral equations in the plane

Local or global existence and uniqueness theorems for nonlinear stochastic functional integral equations are proved. The proofs are based on the successive approximations methods. The formulation includes retarded arguments and hereditary Volterra terms.

A Semigroup Approach to W1,∞-Solutions to a Class of Quasilinear Hyperbolic Systems

In this paper we present a local existence and uniqueness theorem for a W 1,∞-solution to a quasilinear first order system based on semigroup theory. We apply abstract theorems on nonautonomous linear evolution equations to the linearized system and solve the quasilinear system by a fixed point argument.

The Einstein equation for signature type changing spacetimes

We give a local existence and uniqueness theorem for solutions of Einstein’s equations with dust energy momentum tensor in the class of m -dimensional, analytic, transverse, signature type changing spacetimes where the initial condi­tions are given on the hypersurface of signature type change. We also prove a similar theorem in the case that the energy momentum tensor represents a scalar field.

Local solution for the Kadomtsev-Petviashvili equation with periodic conditions

In this article we prove a local existence and uniqueness theorem for the Kadomtsev-Petviashvili Equation (u t +u xxx +uu x ) x −u yy =0) in the Sobolev spaces of orders≥3, with initial values in the same spaces, and periodic boundary conditions. This theorem improves previous results based upon the application of singular perturbation techniques.

Existence and uniqueness of solutionsto volterra's population equation with diffusion and noise

We develop a model for the time evolution of a spatially distributed population and arrive at a stochastic partial differential equation. For a corresponding Neumann problem general local and global existence and uniqueness theorems are proved. Sufficient conditions for the non-negativity of the solution are given A useful comparison theorem is proved

Successive approximations to solutions of stochastic differential equations

In the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of Ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions. We note that Lemma 3 in this paper is a generalization of Gard's lemma and guarantees the existence of functions which satisfy the conditions of Theorems 2 and 3 in this paper, respectively. Theorem 3 includes as a special case a generalization of Yamada's theorem which is proved by the method of the successive approximations.

Hamilton-Jacobi Equations With Singular Boundary Conditions on a free Boundary and Applications to Differential Games

A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games.