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This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of $\mathbb{R}^{n}. $ We show that, under suitable conditions on the initial data, the solution exists globally in time. |
The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and the we recall how to obtain the corresponding kinetic fo... |
Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical simulations. |
Tetrahedral frame fields have applications to certain classes of nematic liquid crystals and frustrated media. We consider the problem of constructing a tetrahedral frame field in three dimensional domains in which the boundary normal vector is included in the frame on the boundary. |
Quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, is proven for $n\geq 3$. This extends a result of [Neu16] in $n=2$. |
A fully parabolic chemotaxis model of Keller-Segel type with local sensing is considered. The system features a signal-dependent asymptotically non-degenerate motility function, which accounts for a repulsion-dominated chemotaxis. |
We prove the possibility of achieving exponentially growing wave propagation in space-time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence o... |
We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semilinear ma... |
We are concerned with the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain $\mathbb{T}\times \mathbb{R}$. For a general initial data settled in Sobolev spaces, we obtain a Lyapunov type instability of the density, the temperature... |
In this paper, we introduce a nonlocal evolution equation inspired by the Córdoba-Córdoba-Fontelos nonlocal transport equation. The Córdoba-Córdoba-Fontelos equation can be regarded as a model for the 2D surface quasigeostrophic equation or the Birkhoff-Rott equation. |
In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two pro... |
We prove the existence of infinitely many nontrivial solutions for time-harmonic nonlinear Maxwell's equations on bounded domains and on $\mathbb{R}^3$ using dual variational methods. In the dual setting we apply a new version of the Symmetric Mountain Pass Theorem that does not require the Palais-Smale condition. |
We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^\lambda \leq(\partial_t -\... |
In a recent paper, Struwe considered the Cauchy problem for a class of nonlinear wave and Scrödinger equations. <br>Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. |
Metal streak artifacts in X-ray computerized tomography (CT) are rigorously characterized here using the notion of the wavefront set from microlocal analysis. The metal artifacts are caused mainly from the mismatch of the forward model of the filtered back-projection; the presence of metallic subjects in an imaging su... |
In this paper we prove the existence of a radial ground state solution for a quasilinear problem involving the mean curvature operator in Minkowski space. |
The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains boundary laye... |
In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$ \begin{cases}-\Delta_{p} u+(\lambda a+1)|u|^{p-2} u=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{u}(u, v), \\ -\Delta_{p} v+(\lambda b+1)|v|^{p-2} v=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{v}(u, v),\end{cases} $$ on l... |
We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson. |
In this paper we study how the (normalised) Gagliardo semi-norms $[u]_{W^{s,p} (\mathbb{R}^n)}$ control translations. In particular, we prove that $\| u(\cdot + y) - u \|_{L^p (\mathbb{R}^n)} \le C [ u ] _{W^{s,p} (\mathbb{R}^n)} |y|^s$ for $n\geq1$, $s \in [0,1]$ and $p \in [1,+\infty]$, where $C$ depends only on $n$... |
We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear in IUMJ (2014)] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. |
Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schrödinger operators are extended to allow singular potentials such as certain $L^p$-functions. The proof is based on accordingly adapted Carleman estimates. |
Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq ... |
We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log $C^{1,1}$ functions. |
The main objective of this paper is twofold. We first show that if the doubly-weighted Bohr spectrum of an almost periodic function exists, then it is either empty or coincides with the Bohr spectrum of that function. |
In this short note we prove a sharp dispersive estimate $\|\mathrm{e}^{\mathrm{i} tH} f\|_\infty < t^{-d/3}\|f\|_1$ for any Cartesian product $\mathbb{Z}^d\mathop\square G_F$ of the integer lattice and a finite graph. This includes the infinite ladder, $k$-strips and infinite cylinders, which can be endowed with ce... |
In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainerman-Sideris [1996_CPAM], and global existence was built by Agemi [200... |
In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda u+h(x,u,\lambda)\,\,\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\... |
Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when $\alpha=1$. For both problems, explicit solutions in terms of the Wright functions are p... |
We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =\Delta u^m - \nabla(\frac{u}{v}\nabla v),&{} x\in\Omega,\ t>0, \medskip v_t =\Delta v -uv,&{}x\in\Omega,\ t>0, \medskip \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial\nu}=0,&{}x\in\partial\Om... |
We establish a weak-strong uniqueness principle for solutions to entropy-dissipating reaction-diffusion equations: As long as a strong solution to the reaction-diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates ... |
In this paper, we constuct the multi-point blowup solutions of self-similar type for the inviscid Burgers equation. The shape and blowup dynamics are precisely described. |
We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. |
We considered the qualitative behavior of the generalization of Einstein's model of Brownian motion when the key parameter of the time interval of \textit{free jump} degenerates. Fluids will be characterized by the number of particles per unit volume (density of fluid) at the point of observation. |
In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t), \end{eqnarray} where $\alpha \in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}... |
We consider the nonlinear eigenvalue problem $[D(u (t))u |
We investigate the evolution of a two-phase viscoelastic material at finite strains. The phase evolution is assumed to be irreversible: One phase accretes in time in its normal direction, at the expense of the other. |
By the use of a new vertical estimate introduced by the authors in the context of relaxation shocks for shallow water flow, we both simplify and extend the basic $L^1\cap H^3$ stability results of Mascia and Zumbrun for viscous shock waves, in particular extending their results for Lax waves to the undercompressive cas... |
In this paper, we establish the existence and uniqueness of subsonic flows with a contact discontinuity in a two-dimensional finitely long slightly curved nozzle by prescribing the entropy, the Bernoulli's quantity and the horizontal mass flux distribution at the entrance and the flow angle at the exit. The proble... |
The purpose of this paper is twofold. First we study bifurcations of connected sets of critical orbits of some invariant functional from a given family of critical orbits. |
In the significant work of [2], Alinhac proved the global existence of small solutions for 2D quasilinear wave equations under the null conditions. The proof heavily relies on the fact that the initial data have compact support [22]. |
In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form $\Delta u + W \cdot \nabla u = 0$ in $\mathbb{R}^2$, where $W = W_1 + i W_2$ with each $W_j$ real-valued. |
This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expa... |
A Rayleigh wave is a type of surface wave that propagates in the boundary of an elastic solid with traction (or Neumann) boundary conditions. Since the 1980s much work has been done on the problem of constructing a leading term in an \emph{approximate} solution to the rather complicated second-order quasilinear hyperb... |
\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, \mu}\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + <br>K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi \in H_\mu^1 %\qquad c\le c_\mu, \end{equation*} in the context of the study o... |
This paper address the approximation of the dynamic of two fluids with non matching densities and viscosities modeled by the Allen-Cahn equation coupled with the time dependent Navier-Stokes equations. Existence, uniqueness and a maximum principle are obtained for a totally implicit semi-discrete in time formulation. |
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sens... |
In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. |
We extend Loeper's $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1 < p < +\infty$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper's $2$-Wasserstein stability estimate and the recent stability e... |
We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - \Delta_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calderón & Zygmund type estimates at the gradient level. |
In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. |
Three inverse boundary value problems for the heat equations in one space dimension are considered. Those three problems are: extracting an unknown interface in a heat conductive material, an unknown boundary in a layered material or a material with a smooth heat conductivity by employing a single set of the temperatu... |
This paper studies the expansion into vacuum of a wedge of gas at rest. This problem catches several important classes of wave interactions in the context of 2D Riemann problems. |
Despite the many applications of rate-independent systems, their regularity theory is still largely unexplored. Usually, only weak solution with potentially very low regularity are considered, which requires non-smooth techniques. |
This paper studies the $d$-dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions. We apply Droniou's approach for non-coercive linear elliptic problems to obtain the existence and uniqueness of the solution of the penalized problem,... |
Inspired by numerical studies of the aggregation equation, we study the effect of regularization on nonlocal interaction energies. We consider energies defined via a repulsive-attractive interaction kernel, regularized by convolution with a mollifier. |
In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter $\beta^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range $0<\beta^2<4\pi$ via the variatio... |
There are two distinct regimes commonly used to model traveling waves in stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of multiple immiscible strata. The former is the more physically accurate descriptio... |
The paper presents a model of a dynamic crack with a wavy surface. So far, theoretical analysis of crack front waves has been performed only for in-plane perturbations of the crack front. |
We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a compact "obstacle". The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic... |
We present an elementary approach to observe frequency cascade on forced nonlinear Schr{ö}dinger equations. The forcing term consists of a constant term, perturbed by a modulated Gaussian well. |
In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in $W^{2,1;p}_{loc}$. |
We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $d\mu_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a F... |
We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap \{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hy... |
A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. |
We generalize the $L^p$ spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. |
We consider a disk-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) |
In this paper we will prove that suitable weak solutions of three dimensional Navier-Stokes equations in bounded domain can be constructed by a particular type of artificial compressibility approximation. |
A general fractional relaxation equation is considered with a convolutional derivative in time introduced by A. Kochubei (Integr. Equ. |
We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit $d_{12}$ of this random variable can be expressed in terms of PDEs related to the Fisher--KPP equation. |
This paper studies some $L^p-L^q$ estimates for the dissipative or conservative Moore-Gibson-Thompson (MGT) equations in the whole space $\mathbb{R}^n$. Our contributions are twofold. |
In this paper we use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation $\mathcal{L}u=\lambda u+|u|^{p}$, where $\mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $\lambda$ is the bifurcation para... |
We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. |
We establish the local well-posedness of the Bartnik static metric extension problem for arbitrary Bartnik data that perturb that of any sphere in a Schwarzschild $\{t=0\}$ slice. Our result in particular includes spheres with arbitrary small mean curvature. |
This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. |
We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form ${\mathcal{E}}$ and $\mu$ is a nonnegative bounded smooth measure with respect to the capaci... |
The goal of this paper is to study the slow motion of solutions of the nonlocal Allen-Cahn equation in a bounded domain $\Omega \subset \mathbb{R}^n$, for $n > 1$. The initial data is assumed to be close to a configuration whose interface separating the states minimizes the surface area (or perimeter); both local a... |
In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors. |
We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation \[iu_t+\Delta u=u(e^{4\pi |u|^2}-1)\] in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mas... |
We show that a necessary condition for $T$ to be a potential blow up time is $\lim\limits_{t\uparrow T}\|v(\cdot,t)\|_{L_3}=\infty$. |
We establish the existence of multi-bump solutions for the following class of quasilinear problems $$ <br>- \Delta_{ p(x) } u + \big( \lambda V(x) + Z(x) \big) u ^{ p(x)-1 } = f(x,u) \text{ in } \mathbb R^N, \, u \ge 0 \text{ in } \mathbb R^N, $$ where the nonlinearity $ f \colon \mathbb R^N \times \mathbb R \to \mathb... |
The paper is devoted to studying the asymptotics of the family $(\mu^\varepsilon)$ of stationary measures of the Markov process generated by the flow of stochastic 2D Navier-Stokes equation with smooth white noise. By using the large deviations techniques, we prove that this family is exponentially tight in $H^{1-\gam... |
The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: <br>$$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~\Omega\times (0,T)\subset \mathbb{R}^{n+1}, <br>$$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any $... |
In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of ... |
This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity $\|\rho_0\|_{L^\infty}+\|b_0\|_{L^3}$ is suitably small and ... |
We study the global existence of the parabolic-parabolic Keller-Segel system in $\R^d , d \ge 2$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. |
In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 & on\quad \partial\Omega\\ \end{array} \right. \end{equation*} and coupled systems \begin{equation*} \left\{\begin{array}... |
The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $\phi$ and the potenti... |
We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac like inequalities. |
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [Arch. Rational Mech. Anal., 43 (1971)] by introducing a suitable P-function. |
We are interested in a WKB analysis of the Logarithmic Non-Linear Schrödinger Equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed.... |
We consider the linear Zakharov-Kuznetsov equation on a rectangle with a left Dirich-let boundary control. Using the flatness approach, we prove the null controllability of this equation and provide a space of analytic reachable states. |
We consider a susceptible, infected, and recovered infectious disease model which incorporates a vaccination rate. In particular, we study the problem of choosing the vaccination rate in order to reduce the number of infected individuals to a given threshold as quickly as possible. |
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. |
In this paper, we consider the inverse boundary value problem for the polyharmonic operator. We prove that the second order perturbations are uniquely determined by the corresponding Dirichlet to Neumann map. |
This paper is concerned with existence results for the singular $p$-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. More precisely, by using variational methods combined with the Mountain pass theorem and the Ekeland variational principle, we establish the existence and multip... |
This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied a randomisation method based on the Laplacian with harmonic potential. Here we extend our previous results to the case of any polynomial and confining potential $V$ on $\mathbb{R}^d$. |
This paper estimates the location and the width of the nodal set of the first Neumann eigenfunctions on a smooth convex domain $\Omega \subset \mathbb R^n$, whose length is normalized to be 1 and whose cross-section is contained in a ball of radius $\epsilon$. In \cite{CJK2009}, an $O(\epsilon)$ bound was obtained by ... |
We generalize the no-neck result of Qing-Tian \cite{QT} to show that there is no neck during blowing up for the $n$-harmonic flow as $t\to\infty$. As an application of the no-neck result, we settle a conjecture of Hungerbühler \cite {Hung} by constructing an example to show that the $n$-harmonic map flow on an $n$-dim... |
We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction. |
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