Local Uniqueness Result Research Articles

Uniqueness and Non-Uniqueness Results for Spacetime Extensions

Abstract Given a function $f: A \to{\mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A \subseteq{\mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${\mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${\mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.

Local minimizers for variational obstacle avoidance on Riemannian manifolds

<abstract><p>This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.</p></abstract>

The diffusion identification in a SIS reaction-diffusion system.

This article is concerned with the determination of the diffusion matrix in the reaction-diffusion mathematical model arising from the spread of an epidemic. The mathematical model that we consider is a susceptible-infected-susceptible model with diffusion, which was deduced by assuming the following hypotheses: The total population can be partitioned into susceptible and infected individuals; a healthy susceptible individual becomes infected through contact with an infected individual; there is no immunity, and infected individuals can become susceptible again; the spread of epidemics arises in a spatially heterogeneous environment; the susceptible and infected individuals implement strategies to avoid each other by staying away. The spread of the dynamics is governed by an initial boundary value problem for a reaction-diffusion system, where the model unknowns are the densities of susceptible and infected individuals and the boundary condition models the fact that there is neither emigration nor immigration through their boundary. The reaction consists of two terms modeling disease transmission and infection recovery, and the diffusion is a space-dependent full diffusion matrix. The determination of the diffusion matrix was conducted by considering that we have experimental data on the infective and susceptible densities at some fixed time and in the overall domain where the population lives. We reformulated the identification problem as an optimal control problem where the cost function is a regularized least squares function. The fundamental contributions of this article are the following: The existence of at least one solution to the optimization problem or, equivalently, the diffusion identification problem; the introduction of first-order necessary optimality conditions; and the necessary conditions that imply a local uniqueness result of the inverse problem. In addition, we considered two numerical examples for the case of parameter identification.

Domain stabilization in probability in a fixed time for nonlinear stochastic systems via feedback control

AbstractThis paper is concerned with the domain stabilization in probability in a fixed time for nonlinear stochastic systems. More specifically, given a target domain , a probability and a fixed time , we aim to find suitable state feedback control laws such that the trajectory of the resulting closed‐loop system, starting from outside , will reach the target domain in the fixed time with the minimum probability . To this end, we prove a local existence and uniqueness result of solutions to nonlinear stochastic systems under mild conditions, and derive upper bounds of the mean of the first hitting time with respect to under various rigorous conditions. By these results, a Lyapunov‐based controller design is given to guarantee the domain stability for nonlinear stochastic systems with affine control inputs. Besides, inequalities of quadratic forms focusing on semi‐linear stochastic systems are used to design a linear state feedback controller for achieving the domain stability in probability. Several examples are given for demonstration.

Qualitative properties of zero-current ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes

<p style='text-indent:20px;'>We consider a one-dimensional Poisson-Nernst-Planck system with two oppositely charged particles and nonuniform finite ion sizes modeled through a local hard-sphere potential. The existence and local uniqueness result is established under the framework of geometric singular perturbation theory. Treating the fi- nite ion size as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to further study the qualitative properties of zero-current ionic flows, a special state among the range of the value for ionic current, which is significant for physiology. Of particular interest are the effects on the zero-current ionic flows from finite ion sizes, diffusion coefficients and ion valences. Critical potentials are identified and their important roles played in the study of ionic flow properties are characterized. Those non-intuitive observations from mathematical analysis of the system provide better understandings of the mechanism of ionic flows through membrane channels, particularly the internal dynamics of ionic flows, which cannot be detected via current technology. Numerical simulations are performed to provide more intuitive illustrations of the analytical results.</p>

A multidimensional diffusion coefficient determination problem for the time-fractional equation

In this paper, we consider a multidimensional inverse problem for a fractional diffusion equation. The inverse problem is reduced to the equivalent integral equation. For solving this equation the Schauder principle is applied. The local existence and uniqueness results are obtained.

LOCAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS FOR A NONLINEAR PSEUDOPARABOLIC EQUATION WITH VISCOELASTIC TERM

In this paper, we investigate an initial boundary value problem for a nonlinear pseudoparabolic equation. At first, by applying the Faedo-Galerkin, we prove local existence and uniqueness results. Next, by constructing Lyapunov functional, we establish a sufficient condition to obtain the global existence and exponential decay of weak solutions.

The number of positive solutions to the Brezis-Nirenberg problem

In this paper we are concerned with the well-known Brezis-Nirenberg problem { − Δ u = u N + 2 N − 2 + ε u , a m p ; in Ω , u > 0 , a m p ; in Ω , u = 0 , a m p ; on ∂ Ω . \begin{equation*} \begin {cases} -\Delta u= u^{\frac {N+2}{N-2}}+\varepsilon u, &{\text {in}~\Omega },\\ u>0, &{\text {in}~\Omega },\\ u=0, &{\text {on}~\partial \Omega }. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small ε > 0 \varepsilon >0 was obtained (see Monica Musso and Angela Pistoia [Indiana Univ. Math. J. 51 (2002), pp. 541–579]). However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small ε > 0 \varepsilon >0 . By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and Green’s function of the domain Ω \Omega and then obtain a type of local uniqueness results of blow-up solutions. Lastly we give a description of the number of positive solutions for small positive ε \varepsilon , which depends also on Green’s function.

A singularly perturbed Kirchhoff problem revisited

In this paper, we revisit the singularly perturbation problem(0.1)−(ϵ2a+ϵb∫R3|∇u|2)Δu+V(x)u=|u|p−1uin R3, where a,b,ϵ>0, 1<p<5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem−(a+b∫R3|∇u|2)Δu+u=|u|p−1uin R3. Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for ϵ>0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity.

Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

AbstractWe show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

Curvature flow with driving force on fixed boundary points

In this paper, we consider the curvature flow with driving force on fixed boundary points in the plane. We give a general local existence and uniqueness result of this problem with \(C^2\) initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$ R N

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent: 0.1 $$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$ For $$N\ge 4$$ and $$\sigma \in (\frac{1}{2}, 1),$$ we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied whenbanddare small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions whenk&gt;0is sufficiently large.

Local uniqueness for an inverse boundary value problem with partial data

In dimension n ≥ 3 n\geq 3 , we prove a local uniqueness result for the potentials q q of the Schrödinger equation − Δ u + q u = 0 -\Delta u+qu=0 from partial boundary data. More precisely, we show that potentials q 1 , q 2 ∈ L ∞ q_1,q_2\in L^\infty with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where q 1 ≥ q 2 q_1\geq q_2 and q 1 ≢ q 2 q_1\not \equiv q_2 .

Pohozaev identities and their applications to nonlinear elliptic equations

Pohozaev identity plays a very important role in proving the existence and nonexistence results for the nonlinear elliptic partial differential equations. In this review, we aim to introduce Pohozaev identity and its applications in some classical elliptic partial differential equations. First, we introduce Pohozaev identities for several typical elliptic equations on bounded domains or unbounded domains, and the necessary conditions for the existence of solutions, by which we get some existence and nonexistence results of the solutions. Next, we will show a local Pohozaev identity and its application in proving the compactness of the sequences of approximate solutions, by which we obtain infinitely many solutions of some typical nonlinear elliptic equations. At last, we also introduce some local uniqueness results of peak solutions for some nonlinear elliptic partial differential equations, which can also be obtained by using the local Pohozaev identity. Using this type of uniqueness result, we can prove that the peak solutions can keep the symmetry that the equations have.

Convergence analysis in near-field imaging for elastic waves

A significant method has recently been developed for solving the inverse elastic surface scattering problem which arises from near-field imaging applications. The method utilizes the transformed field expansion along with the Fourier series expansion to deduce an analytic solution for the direct problem. Implemented via the fast Fourier transform, an explicit reconstruction formula is obtained to solve the linearized inverse problem. Numerical examples show that the method is efficient and effective to reconstruct scattering surfaces with subwavelength resolution. This paper is devoted to the mathematical analysis of the proposed method. The well-posedness is established for the solution of the direct problem. The convergence of the power series solution is examined. A local uniqueness result is proved for the inverse problem where a single incident field with a fixed frequency is needed. The error estimate is derived for the reconstruction formula. It provides a deep insight on the trade-off among resolution, accuracy, and stability of the solution for the inverse problem.

On the prescribed scalar curvature problem in [formula omitted], local uniqueness and periodicity

We obtain a local uniqueness result for bubbling solutions of the prescribed scalar curvature problem in RN. Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y). In particular, we prove in this paper that if K(y) is periodic in y1 with period 1 and has a local maximum at 0, then a bubbling solution whose blow-up set is {(jL,0,⋯,0):j=0,±1,±2,⋯} must be periodic in y1 provided the positive integer L is large enough.

Convergence analysis in near-field imaging

This paper is devoted to the mathematical analysis of the direct and inverse modeling of the diffraction by a perfectly conducting grating surface in the near-field regime. It is motivated by our effort to analyze recent significant numerical results, in order to solve a class of inverse rough surface scattering problems in near-field imaging. In a model problem, the diffractive grating surface is assumed to be a small and smooth deformation of a plane surface. On the basis of the variational method, the direct problem is shown to have a unique weak solution. An analytical solution is introduced as a convergent power series in the deformation parameter by using the transformed field and Fourier series expansions. A local uniqueness result is proved for the inverse problem where only a single incident field is needed. On the basis of the analytic solution of the direct problem, an explicit reconstruction formula is presented for recovering the grating surface function with resolution beyond the Rayleigh criterion. Error estimates for the reconstructed grating surface are established with fully revealed dependence on such quantities as the surface deformation parameter, measurement distance, noise level of the scattering data, and regularity of the exact grating surface function.

Well-posedness for the stochastic higher order modified Camassa–Holm equation

In this article, the stochastic higher order modified Camassa–Holm equation with additive noise is considered. Using the Fourier restriction norm method and bilinear estimates, the local existence and uniqueness result is proved by the contraction mapping principle. The global existence of the H1(R) solution of this problem is also obtained by energy inequality.

On the determination of a function from an elliptical Radon transform

In recent years, Radon-type transforms that integrate functions over various sets of ellipses/ellipsoids have been considered in synthetic aperture radar, ultrasound reflection tomography, and radio tomography. In this paper, we consider the transform that integrates a given function in Rn over a set of solid ellipses (when n=2) or solid ellipsoids of rotation (when n⩾3) with a fixed eccentricity and foci restricted to a hyperplane. Inversion formulas are obtained for appropriate classes of functions that are even with respect to the hyperplane. Stability estimates and local uniqueness results are also provided.