Local Uniqueness Result Research Articles

Generalized Hunter–Saxton Equations, Optimal Information Transport, and Factorization of Diffeomorphisms

We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher’s information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalizations of the μ-Hunter–Saxton equation, used to model liquid crystals under the influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray.The descending property of the metrics is used to obtain a novel factorization of diffeomorphisms. Analogous to the polar factorization in optimal mass transport, this factorization solves an optimal information transport problem. It can be seen as an infinite-dimensional version of QR factorization of matrices.

Point Measurements for a Neumann-to-Dirichlet Map and the Calderón Problem in the Plane

This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded, and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite signs are applied at the very same points, uniquely determine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweep data extend as a holomorphic function of two variables to some (interior) neighborhood of the product boundary. It follows that the whole Neumann-to-Dirichlet map is characterized by the derivatives of the bisweep data at an arbitrary point. On the diagonal of the product boundary, these derivatives can be given with the help of the derivatives of the (relative) boundary potentials at some fixed point caused by the distributional current densities supported at the same point, and thus such point measurements uniquely define the Neumann-to-Dirichlet map. This observation also leads to a new, truly local uniqueness result for the so-called Calderón inverse conductivity problem.

A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity

We prove the existence and uniqueness of solution to a one-dimensional hyperbolic–parabolic system arising in the study of magneto-viscoelasticity. Specifically, the local existence and uniqueness result is proved on application of the fixed point theorem; then, a uniform a priori estimate of the solution is established. The latter, via a continuation method, allows as to obtain the global result. A crucial tool to achieve such a result is a technical lemma concerning the only viscoelastic contribution; it relies on the assumptions that the memory kernel is positive, monotonically non-increasing and convex.

Determining Planar Multiple Sound-Soft Obstacles from Scattered Acoustic Fields

An inverse problem is considered where the structure of multiple sound-soft planar obstacles is to be determined given the direction of the incoming acoustic field and knowledge of the corresponding total field on a curve located outside the obstacles. A local uniqueness result is given for this inverse problem suggesting that the reconstruction can be achieved by a single incident wave. A numerical procedure based on the concept of the topological derivative of an associated cost functional is used to produce images of the obstacles. No a priori assumption about the number of obstacles present is needed. Numerical results are included showing that accurate reconstructions can be obtained and that the proposed method is capable of finding both the shapes and the number of obstacles with one or a few incident waves.

Direct and inverse problems related to MEMS

This paper deals with direct and inverse evolution problems which come up in studying micro-electro-mechanical-systems: here we consider a nonlinear and nonlocal MEMS model. The inverse problem consists of recovering a time-varying Coulomb potential by exploiting some accessible measurements, which depend on the dynamic displacement of the system. Local existence, uniqueness and continuous dependence results are proved for both direct and inverse problems.

On convergence of Lavrentiev regularization for semilinear parabolic optimal control problems with pointwise control and state constraints

AbstractWe consider Lavrentiev regularization for a class of semilinear parabolic optimal control problems with control constraints and pointwise state constraints and review convergence results for local solutions under Slater type assumptions as well as quadratic growth conditions. Moreover, we state a local uniqueness result for local optima under the assumptions of strict separability of the active sets as well as a second order sufficient condition for the regularized solution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crystalline Mean Curvature Flow of Convex Sets

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in Open image in new window. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.

Remarks on Nonlinear Elliptic Problems on Long Domains

Abstract In this paper, we discuss the behaviour of solutions near a turning point for weakly nonlinear elliptic equations on long domains. We are looking for results independent of the domain length. In the process we obtain some local uniqueness and non-degeneracy results for homoclinic solutions on infinite strips.

On a mixed finite element formulation of a second‐order quasilinear problem in the plane

AbstractWe analyze a mixed finite element discretization of a second‐order quasilinear problem based on the Raviart‐Thomas space. We prove that the discrete problem is solvable and provide a local uniqueness result for the solution. We also obtain optimal order L2‐error estimates for both the scalar variable and the associated flux. The main feature of our method is that it is free from the boundness conditions required in previous works on the coefficients of the quasilinear operator. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 90–103, 2004.

Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.

Slow motion and metastability for a nonlocal evolution equation

In this paper we consider a nonlocal evolution equation in one dimension, which describes the dynamics of a ferromagnetic system in the mean field approximation. In the presence of a small magnetic field, it admits two stationary and homogeneous solutions, representing the stable and metastable phases of the physical system. We prove the existence of an invariant, one dimensional manifold connecting the stable and metastable phases. This is the unstable manifold of a distinguished, spatially nonhomogeneous, stationary solution, called the critical droplet.(4, 10) We show that the points on the manifold are droplets longer or shorter than the critical one, and that their motion is very slow in agreement with the theory of metastable patterns. We also obtain a new proof of the existence of the critical droplet, which is supplied with a local uniqueness result.

Weyl–Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

Branched spines and contact structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields.

An Euler-Poisson system in plasmas

AbstractA local existence and uniqueness result is proved for the three-dimensional Euler-Poisson system without a pressure term which arises in plasma physics.

Inverse obstacle problem: local uniqueness for rougher obstacles and the identification of a ball

In this paper, we study the determination of the shape of an obstacle with rough boundary (not necessary Lipschitz) from its scattering amplitude. Following a technique used by Liu and Nachman, we obtain extensions of the uniqueness of the k-core and the local uniqueness result for Lipschitz obstacles to that of rougher ones. Using the uniqueness of the k-core, we prove that a ball in is identifiable from two linearly independent incident directions.

An inverse problem for the linear viscoelastic Kirchhoff plate

We consider the linear Kirchhoff plate model subject to a viscoelastic damping of integral type. The damping term contains a time-dependent convolution kernel accounting for the long-range memory effects. The mechanical behavior of the plate is influenced by this memory function which is a priori unknown. That leads us to consider the question of identifying it. We study this kind of inverse problem by using the integrodifferential evolution equation governing the vertical deflection of the plate and coupling it with a set of overposed initial and boundary conditions. The problem obtained is then reduced to a nonlinear Volterra integral equation of second kind for the unknown memory kernel. Then, via the Contraction Principle, we prove local (in time) existence and uniqueness results. In addition, we show the Lipschitz continuous dependence upon the data. These results also apply to a viscoelastic beam model.

An identification problem for the Maxwell equations in a non-homogeneous dispersive medium

We consider the propagation of electromagnetic waves in a non-homogeneous medium. The related constitutive relations contain time and space dependent convolution kernels. Since they are a priori unknown, a basic question concerns their identification. In the present paper, this is obtained by reducing the problem to a system of nonlinear integral equations of the second kind. Via a Contraction Theorem, we prove local (in time) existence and uniqueness results. Lipschitz continuous dependence upon the data is also proved.

Existence of standing pulse solutions for an excitable activator-inhibitory system

Invariant manifold techniques are used to establish the existence of a standing wave solution for a system of reaction-diffusion equations modeled after excitable media. As a by-product of the method of proof, we obtain a local uniqueness result.

An analysis of facilitated-diffusion problems

This paper develops the mathematical analysis of nonlinear boundary-value problems that model the facilitated diffusion of a substrate across a membrane. The analysis includes bounds for solutions, global existence results, local uniqueness and continuity results, and estimates for parameter ranges for uniqueness of solutions.

A perturbation study of a jet-like annular free boundary problem and an application to an optimal control problem

The NASH-MOSER Generalized Implicit Function Theorem is applied to an annular Free Boundary Problem related to jet or wave phenomena : local existence, uniqueness, and data dependence results are derived, together with an application to an Optimal Control Problem.