Search results for "uniqueness"
showing 10 items of 211 documents
The Neumann Problem for the Total Variation Flow
2004
This chapter is devoted to prove existence and uniqueness of solutions for the minimizing total variation flow with Neumann boundary conditions, namely $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right) in Q = (0,\infty ) \times \Omega , \hfill \\ \frac{{\partial u}} {{\partial \eta }} = 0 on S = (0,\infty ) \times \partial \Omega , \hfill \\ u(0,x) = u_0 (x) in x \in \Omega , \hfill \\ \end{gathered} \right. $$ (2.1) where Ω is a bounded set in ℝ N with Lipschitz continuous boundary ∂ Ω and u0 ∈ L1(Ω). As we saw in the previous chapter, this partial differential equation appears when one uses the steepest descent method …
Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting
2018
We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a L\'evy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we consider BSDEs where the Poisson random measure admits only jumps of size larger than $1/n$. We show con…
Spatial Competition in Quality
2011
We develop a model of vertical innovation in which firms incur a market entry cost and position themselves in the quality space. Once established, firms compete monopolistically, selling to consumers with heterogeneous tastes for quality. We establish the general existence and conditional uniqueness of the pricing game in such vertically differentiated markets with a potentially large number of active firms. Turning to firms’ entry decisions, exogenously growing productivities induce firms to enter the market sequentially at the top end of the quality spectrum. We spell out the conditions under which the entry problem is replicated over time so that each new entrant improves incumbent quali…
Uniqueness of solutions for some elliptic equations with a quadratic gradient term
2008
We study a comparison principle and uniqueness of positive solutions for the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with lower order terms. A model example is given by −Δu + λ |∇u| 2 u r = f (x) ,λ , r >0. The main feature of these equations consists in having a quadratic gradient term in which singularities are allowed. The arguments employed here also work to deal with equations having lack of ellipticity or some dependence on u in the right hand side. Furthermore, they could be applied to obtain uniqueness results for nonlinear equations having the p-Laplacian operator as the principal part. Our results improve those already known, even…
The 1-Harmonic Flow with Values into $\mathbb S^{1}$
2013
We introduce a notion of solution for the $1$-harmonic flow, i.e., the formal gradient flow of the total variation functional with respect to the $L^2$-distance, from a domain of $\mathbb R^m$ into a geodesically convex subset of an $N$-sphere. For such a notion, under homogeneous Neumann boundary conditions, we prove both existence and uniqueness of solutions when the target space is a semicircle and the existence of solutions when the target space is a circle and the initial datum has no jumps of an “angle” larger than $\pi$. Earlier results in [J. W. Barrett, X. Feng, and A. Prohl, SIAM J. Math. Anal., 40 (2008), pp. 1471--1498] and [X. Feng, Calc. Var. Partial Differential Equations, 37…
Uniqueness of positive multi-lump bound states of nonlinear Schr�dinger equations
2003
In this paper we are concerned with multi-lump bound states of the nonlinear Schrodinger equation
Cooperative compressive power spectrum estimation in wireless fading channels
2017
This paper considers multiple wireless sensors that cooperatively estimate the power spectrum of the signals received from several sources. We extend our previous work on cooperative compressive power spectrum estimation to accommodate the scenario where the statistics of the fading channels experienced by different sensors are different. The signals received from the sources are assumed to be time-domain wide-sense stationary processes. Multiple sensors are organized into several groups, where each group estimates a different subset of lags of the temporal correlation. A fusion centre (FC) combines these estimates to obtain the power spectrum. As each sensor group computes correlation esti…
Common fixed points in cone metric spaces for CJM-pairs
2011
Abstract In this paper we introduce some contractive conditions of Meir–Keeler type for two mappings, called f - M K -pair mappings and f - C J M -pair (from Ciric, Jachymski, and Matkowski) mappings, in the framework of regular cone metric spaces and we prove theorems which guarantee the existence and uniqueness of common fixed points. We give also a fixed point result for a multivalued mapping that satisfies a contractive condition of Meir–Keeler type. These results extend and generalize some recent results from the literature. To conclude the paper, we extend our main result to non-regular cone metric spaces by using the scalarization method of Du.
Stochastic equation of population dynamics with diffusion on a domain
2003
We consider Lotka-Volterra competition model with diffusion in a territorial domain with a stochastic perturbation which represents the random variations of environment conditions. We prove the existence, the uniqueness and the positivity of the solution. Moreover, the stochastic boundedness of the solution is analized.
Numerical approach to the exact controllability of hyperbolic systems
2005
In this paper we present the numerical implementation of H.U.M. (Hilbert Uniqueness Method, J.L.Lions[1]). We restrict ourselves to the exact boundary controllability of the wave equation, with Dirichlet controls, but the numerical method presented here can be applied to other kinds of controllability. The problem is discretized by a finite elements of first order in space and by a discrete time Galerkin approximation (Dupont [1]). The efficiency of the method is illustrated by numerical results.