On Local Borg-Marchenko Uniqueness Results

We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schr\"odinger operators H_j = -\f{d^2}{dx^2} + q_j, j=1,2 in $L^2 ((0,R))$, $0<R\leq \infty$, are exponentially close, that is, |m_1(z)- m_2(z)| \underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a}), 0<a<R, then $q_1 = q_2$ a.e.~on $[0,a]$. The result applies to any boundary conditions at x=0 and x=R and should be considered a local version of the celebrated Borg-Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrix-valued Schr\"odinger operators.Comment: LaTeX, 18 page

Some uniqueness results for dynamical horizons

We first show that the intrinsic, geometrical structure of a dynamical horizon is unique. A number of physically interesting constraints are then established on the location of trapped and marginally trapped surfaces in the vicinity of any dynamical horizon. These restrictions are used to prove several uniqueness theorems for dynamical horizons. Ramifications of some of these results to numerical simulations of black hole spacetimes are discussed. Finally several expectations on the interplay between isometries and dynamical horizons are shown to be borne out.Comment: 26 pages, 4 figures, v4: references updated, minor corrections, to appear in Advances in Theoretical and Mathematical Physic

Uniqueness results for Zakharov-Kuznetsov equation

In this paper we study uniqueness properties of solutions to the Zakharov-Kuznetsov equation of plasma physic. Given two sufficiently regular solutions $u_1, u_2,$ we prove that, if $u_1-u_2$ decays fast enough at two distinct times, then $u_1\equiv u_2.$Comment: 33 page

Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts

Existence and uniqueness results for a nonlinear stationary system

We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves $L^1$ coupling terms we use the techniques of renormalized solutions for elliptic equations with $L^1$ data. We also prove partial uniqueness results

Some uniqueness results related to the Br\"{u}ck Conjecture

Let f be a non-constant meromorphic function and a = a(z) be a small function of f. Under certain essential conditions, we obtained similar type conclusion of Bruck Conjecture, when f and its differential polynomial P[f] shares a with weight l. Our result improves and generalizes a recent result of Li, Yang, and Liu.Comment: 11 page