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The operator square root of the Laplacian $(-\lap)^{1/2}$ can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems. | An extension problem related to the fractional Laplacian | 11,600 |
We present a two-dimensional (2D) mathematical model of a highly concentrated suspension or a thin film of the rigid inclusions in an incompressible Newtonian fluid. Our objectives are two-fold: (i) to obtain all singular terms in the asymptotics of the overall viscous dissipation rate as the interparticle distance parameter $\delta$ tends to zero, (ii) to obtain a qualitative description of a microflow between neighboring inclusions in the suspension. | Fictitious Fluid Approach and Anomalous Blow-up of the Dissipation Rate
in a 2D Model of Concentrated Suspensions | 11,601 |
We prove wellposedness of the Cauchy problem for the cubic nonlinear Schrodinger equation with Dirichlet boundary conditions and radial data on 3D balls. The main argument is based on a bilinear eigenfunction estimate and the use of $X^{s,b}$ spaces. The last part presents a first attempt to study the non radial case. We prove bilinear estimates for the linear Schrodinger flow with particular initial data. | Cubic nonlinear Schrodinger equation on three dimensional balls with
radial data | 11,602 |
We show that the time evolution of the operator $H = -\Delta + i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance. | Strichartz and Smoothing Estimates for Schroedinger Operators with Large
Magnetic Potentials in R^3 | 11,603 |
Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon = \beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity. In particular, we show that the topological free boundary $\partial\{u>0\}$ can be decomposed into an {\em open} regular set (relative to $\partial\{u>0\}$) which is locally a surface with H\"older-continuous space normal, and a closed singular set. Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems. | A parabolic free boundary problem with Bernoulli type condition on the
free boundary | 11,604 |
The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$. | Global well-posedness for the Schroedinger equation coupled to a
nonlinear oscillator | 11,605 |
The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\times Y$). The caustic set $\Sigma(C)$ of the canonical relation $C$ is characterized as the set of points where the rank of the projection $\pi:C\to X\times Y$ is smaller than its maximal value, $dim(X\times Y)-1$. We derive the $L\sp p(Y)\to L\sp q(X)$ estimates on Fourier integral operators with caustics of corank 1 (such as caustics of type $A\sb{m+1}$, $m\in\N$). For the values of $p$ and $q$ outside of certain neighborhood of the line of duality, $q=p'$, the $L\sp p\to L\sp q$ estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics. | $L\sp p$-$L\sp q$ regularity of Fourier integral operators with caustics | 11,606 |
We obtain optimal continuity in Sobolev spaces for the Fourier integral operators associated to singular canonical relations, when one of the two projections is a Whitney fold. The regularity depends on the type, $k$, of the other projection from the canonical relation ($k=1$ for a Whitney fold). We prove that one loses $(4+\frac{2}{k})^{-1}$ of a derivative in the regularity properties. The proof is based on the $L^2$ estimates for oscillatory integral operators. | Optimal regularity of Fourier integral operators with one-sided folds | 11,607 |
The nonlinear Schroedinger equation possesses three distinct six-parameter families of complex-valued quasi-periodic travelling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x-ct for some real c. In this paper we investigate the stability of the small amplitude travelling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude travelling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability. | Stability of small periodic waves for the nonlinear Schroedinger
equation | 11,608 |
The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile. | Orbital stability of periodic waves for the nonlinear Schroedinger
equation | 11,609 |
This memoir attempts at a systematic study of convergence to stationary state for certain classes of degenerate diffusive equations, by means of well-chosen Lyapunov functionals. Typical examples are the kinetic Fokker--Planck and Boltzmann equation. Many open problems and possible directions for future research are discussed. | Hypocoercivity | 11,610 |
We prove a regularity result for the Poisson problem $-\Delta u = f$, $u |\_{\pa \PP} = g$ on a polyhedral domain $\PP \subset \RR^3$ using the \BK\ spaces $\Kond{m}{a}(\PP)$. These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of $\Kond{m}{a}$--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in $\Kond{m}{a}(\PP)$. | Weighted Sobolev spaces and regularity for polyhedral domains | 11,611 |
T. Riviere proved an energy quantization for Yang-Mills fields defined on n-dimensional Riemannian manifolds, when $n$ is larger than the critical dimension 4. More precisely, he proved that the defect measure of a weakly converging sequence of Yang-Mills fields is quantized, provided the $W^{2,1}$ norm of their curvature is uniformly bounded. In the present paper, we prove a similar quantization phenomenon for the Yamabe problem in a bounded domain $\Omega$ of $R^n$. | Energy Quantization for Yamabe's problem in Conformal Dimension | 11,612 |
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p-Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2), this condition involves comparison of both the functions and of their gradients. | A Liouville-type theorem for the p-Laplacian with potential term | 11,613 |
We extend results of Dos Santos Ferreira-Kenig-Sjoestrand-Uhlmann (math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schroedinger operator determine uniquely the magnetic field related to a Hoelder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights. | Determining nonsmooth first order terms from partial boundary
measurements | 11,614 |
We derive damping estimates and asymptotics of $L^p$ operator norms for oscillatory integral operators with finite type singularities. The methods are based on incorporating finite type conditions into $L^2$ almost orthogonality technique of Cotlar-Stein. | Damping estimates for oscillatory integral operators with finite type
singularities | 11,615 |
We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\"{o}dinger equations. Schr\"{o}dinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. | Soliton solutions for quasilinear Schrödinger equations involving
supercritical exponent in $R^N$ | 11,616 |
We establish that the initial value problem for the quadratic non-linear Schr\"odinger equation $$ iu_t - \Delta u = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this problem is $s_c=-1$ and previous work in \cite{c1} established local well-posedness for $s > -3/4$. | Low regularity solutions for a 2D quadratic non-linear Schrödinger
equation | 11,617 |
We consider the equation $- \Delta u+V(x)u- k(\Del(|u|^{2}))u=g(x,u), u>0, x \in {\BR}^2,$ where $V:{\BR}^2\to {\BR}$ and $g:{\BR}^2 \times {\BR}\to {\BR}$ are two continuous $1-$periodic functions. Also, we assume $g$ behaves like $\exp (\beta |u|^4)$ as $|u|\to \infty.$ We prove the existence of at least one weak solution $u \in H^1({\BR}^2)$ with $u^2 \in H^1({\BR}^2).$ Mountain pass in a suitable Orlicz space together with Moser-Trudinger are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\"{o}dinger equations. Schr\"{o}dinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. | On a class of periodic quasilinear Schrödinger equations involving
critical growth in ${\BR}^2$ | 11,618 |
Mountain pass in a suitable Orlicz space is employed to prove the existence of soliton solutions for a quasilinear Schr\"{o}dinger equation involving critical exponent in ${\BR}^N$. These equations contain strongly singular nonlinearities which include derivatives of the second order. Such equations have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. | Existence of soliton solutions for a quasilinear Schrödinger
equation involving critical exponent in ${\BR}^N$} | 11,619 |
We study the existence of positive radially symmetric solution for the singular $p$-Laplacian Dirichlet problem, $-\bigtriangleup_p u =\lambda |u|^{p-2} u-\gamma u^{-\alpha}$ where $\lambda>0,\gamma>0$ and, $0<\alpha<1$, are parameters and $\Omega$, the domain of the equation, is a ball in $\mathbb{R}^N$. By using some variational methods we show that, if $\lambda$ is contained in some interval, then the problem has a radially symmetric positive solution on the ball. Moreover, we obtain a nonexistence result, whenever $\lambda \leq 0, \gamma<0$ and $\Omega$ is a bounded domain, with smooth boundary. | Existence and nonexistence of solutions for a singular $p$-Laplacian
Dirichlet problem | 11,620 |
We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+ a(x)= \mu u|u|^\frac4{n-2} \hbox{} \mu >0, x\in \Omega \subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial \Omega$$ under weak regularity assumptions on the potential $a(x)$. More precisely in dimension $n\geq 5$ we assume that: \begin{enumerate} \item $a(x)$ belongs to the Lorentz space $L^{\frac n2, d}(\Omega)$ for some $1\leq d <\infty$, \item $a(x) \leq M<\infty \hbox{a.e.} x\in \Omega$, \item the set $\{x\in \Omega|a(x)<0\}$ has positive measure, \item there exists $c>0$ such that $$\int_\Omega (|\nabla u|^2 + a(x) |u|^2) \hbox{} dx \geq c\int_\Omega |\nabla u|^2 \hbox{} dx \hbox{} \forall u\in H^1_0(\Omega).$$ \end{enumerate} \noindent In dimension $n=4$ the hypothesis $(2)$ above is replaced by $$a(x)\leq 0 \hbox{} a.e. \hbox{} x\in \Omega.$$ | On the Yamabe equation with rough potentials | 11,621 |
Motivated by physical and numerical observations of time oscillatory ``galloping'', ``spinning'', and ``cellular'' instabilities of detonation waves, we study Poincar\'e--Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves. | Galloping instability of viscous shock waves | 11,622 |
Global classical solutions to the viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions are shown to converge to zero at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to zero is also obtained in one space dimension but the rate depends on a and differs from that of the linear heat equation. Finally, if 0 < p < 1 and a < 0, finite time extinction occurs for non-negative solutions. | Decay Estimates for a Viscous Hamilton-Jacobi Equation with Homogenious
Dirichet Boundary Conditions | 11,623 |
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo- acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in Rn surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface. | Explicit inversion formulas for the spherical mean Radon transform | 11,624 |
We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast at infinity. | On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger
equations with potential | 11,625 |
This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. | The Fourier Singular Complement Method for the Poisson problem. Part I:
prismatic domains | 11,626 |
This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. | The Fourier Singular Complement Method for the Poisson problem. Part II:
axisymmetric domains | 11,627 |
We analyse a reduced 1D Vlasov--Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the field is split in two terms: an electrostatic field obtained from Poisson's equation and a vector potential term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson and wave equations are due to the coupling with the Vlasov equation through the charge density. We show global existence of weak solutions in the non-relativistic case, and global existence of characteristic solutions in the quasi-relativistic case. Moreover, these solutions are uniquely characterised as fixed points of a certain operator. We also find a global energy functional for the system allowing us to obtain $L^p$-nonlinear stability of some particular equilibria in the periodic setting. | Global Solutions for the One-Dimensional Vlasov-Maxwell System for
Laser-Plasma Interaction | 11,628 |
We study the permanent regimes of the reduced Vlasov-Maxwell system for laser-plasma interaction. A non-relativistic and two different relativistic models are investigated. We prove the existence of solutions where the distribution function is Boltzmannian and the electromagnetic variables are time-harmonic and circularly polarized. | On the harmonic Boltzmannian waves in laser-plasma interaction | 11,629 |
We study the extinction behavior of solutions to the fast diffusion equation $u_t = \Delta u^m$ on $\R^N\times (0,T)$, in the range of exponents $m \in (0, \frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in between two Barenblatt solutions vanishing at time $T$, then the vanishing behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions $u$ with initial data $u_0 = B (1 + o(1))$, near $ |x| >> 1$, which live longer than $B$ and change behaviour at $T$. The behavior of such solutions is governed by $B(\cdot,t)$ up to $T$, while for $t >T$ the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow ($m = \frac{N-2}{N+2}$) the above means that these solutions $u$ develop a singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$ they immediately smoothen up and exhibit the vanishing profile of a sphere. In the appendix we show how to remove the assumption on the bound on $u_0$ from below by a Barenblatt. | On the extinction profile of solutions to fast-diffusion | 11,630 |
Estimates are obtained for the expected volume of intersection of independent Wiener sausages in Euclidean space in the small time limit. The asymptotic behaviour of the weighted diagonal heat kernel norm on compact Riemannian manifolds with smooth boundary is obtained in the small time limit | Expected volume of intersection of Wiener sausages and heat kernel norms
on compact Riemannian manifolds with boundary | 11,631 |
In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds. | The Dirichlet problem for singular fully nonlinear operators | 11,632 |
The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic and homogenous. In particular we prove maximum and comparison principle, Holder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators. | Eigenvalue, maximum principle and regularity for fully non linear
homogeneous operators | 11,633 |
We consider non-linear elliptic equations having a measure in the right hand side, of the type $ \divo a(x,Du)=\mu, $ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calder\'on-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates. | The Calderón-Zygmund theory for elliptic problems with measure data | 11,634 |
We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schr\"odinger (NLS) equation $iu_t + \Delta u = \mu |u|^{4/d} u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L^2_x(\R^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in dimensions $d\geq 3$ for the mass-critical NLS and by Kenig and Merle, \cite{merlekenig}, in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in $L^2_x(\R^d)$ for the defocusing NLS in three and higher dimensions with spherically symmetric data. | Minimal-mass blowup solutions of the mass-critical NLS | 11,635 |
We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schr\"odinger equation $iu_t + \Delta u = |u|^{4/n} u$ for large spherically symmetric $L^2_x(\R^n)$ initial data in dimensions $n\geq 3$. After using the reductions in \cite{compact} to reduce to eliminating blowup solutions which are almost periodic modulo scaling, we obtain a frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to \cite{ckstt:gwp}, \cite{RV}, \cite{thesis:art}) in order to conclude the argument. | Global well-posedness and scattering for the mass-critical nonlinear
Schrödinger equation for radial data in high dimensions | 11,636 |
In this paper the author studies the problem of the homogenization of a diffusion perturbed by a periodic reflection invariant vector field. The vector field is assumed to have fixed direction but varying amplitude. The existence of a homogenized limit is proven and formulas for the effective diffusion constant are given. In dimension $d=1$ the effective diffusion constant is always less than the constant for the pure diffusion. In $d>1$ this property no longer holds in general. | On homogenization of a diffusion perturbed by a periodic reflection
invariant vector field | 11,637 |
In the present paper, we consider a class of two players infinite horizon differential games, with piecewise smooth costs exponentially discounted in time. Through the analysis of the value functions, we study in which cases it is possible to establish the existence Nash equilibrium solutions in feedback form. We also provide examples of piecewise linear costs whose corresponding games have either infinitely many Nash equilibria or no solutions at all. | Infinite Horizon Noncooperative Differential Games with Non-Smooth Costs | 11,638 |
The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a rescaled self-similar solution of the linear heat equation is shown, the rescaling factor being $(\log(t))^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables. | Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical
exponent | 11,639 |
We show that Kruzhkov's theory of entropy solutions to multidimensional scalar conservation laws can be entirely recast in L2 and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by Giga, Miyakawa, Osher, Tsai and the author. | L2 formulation of multidimensional scalar conservation laws | 11,640 |
We consider an operator $ P $ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $ \sigma $ is not constant on $ \Char P $. Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for $ P $ consists of closed curves in a ring-shaped open set around the origin. We prove that then $ P $ is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for $ P $ is $ G^{k+1} $ and that this is sharp. | Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for
Sums of Squares of Vector Fields | 11,641 |
We present an elementary, $L^2,$ proof of Fedi\u{\i}'s theorem on arbitrary (e.g., infinite order) degeneracy and extensions. In particular, the proof allows and shows $C^\infty,$ Gevrey, and real analytic hypoellipticity, and allows the coefficents to depend on the remaining variable as well. | An elementary proof of Fediĭ's theorem and extensions | 11,642 |
In 2001, H. Koch and D. Tataru proved the existence of global in time solutions to the incompressible Navier-Stokes equations in ${\mathbb{R}}^d$ for initial data small enough in $BMO^{-1}$. We show in this article that the Koch and Tataru solution has higher regularity. As a consequence, we get a decay estimate in time for any space derivative, and space analyticity of the solution. Also as an application of our regularity theorem, we prove a regularity result for self-similar solutions. | Regularity of solutions to the Navier-Stokes equations evolving from
small data in BMO^{-1} | 11,643 |
In this paper we establish the existence of extremal functions for weighted functionals with critical exponential growth in R^2, which arise from Henon-type equations. The proof is based on the notion of spherical symmetrization with respect to a measure, which allows us to reduce the problem to a one dimensional functional as in the proof due to Carleson and Chang for the unweighted case. | On the existence of maximizers for functionals with critical exponential
growth in R^2 | 11,644 |
We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevi\v{c} operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey hypoellipticity theorem analogous to our recent result for the corresponding Oleinik-Radkevi\v{c} operator. | A Class of Sums of Squares with a Given Poisson-Treves Stratification | 11,645 |
In an interesting note, E.M. Stein observed some 20 years ago that while the Kohn Laplacian $\square_b$ on functions is neither locally solvable nor (analytic) hypoelliptic, the addition of a non-zero complex constant reversed these conclusions at least on the Heisenberg group, and Kwon reproved and generalized this result using the method of concatenations. Recently Hanges and Cordaro have studied this situation on the Heisenberg group in detail. Here we give a purely $L^2$ proof of Stein's result using the author's now classical construction of $(T^p)_\phi = \phi T^p +...,$ where $T$ is the 'missing direction' on the Heisenberg group. | Analytic hypoellipticity for $\square_b + c$ on the Heisenberg group: an
$L^2$ approach | 11,646 |
In this paper we are interested in constructing WKB approximations for the non linear cubic Schr\"odinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation. | The WKB method and geometric instability for non linear Schrodinger
equations on surfaces | 11,647 |
Using variational methods, we construct approximate solutions for the Gross-Pitaevski equation which concentrate on circles in $\R^3$. These solutions will help to show that the $L^2$ flow is unstable for the usual topology and for the projective distance. | Geometric and projective instability for the Gross-Pitaevski equation | 11,648 |
We construct N-harmonic functions in a domain with one isolated singularity on the boundary of the domain. By using solutions of the spherical p-harmonic spectral problem, we give an inductive method to produce a large variety of separable p-harmonic functions. | Boundary singularities of N -harmonic functions | 11,649 |
The 3D spatially periodic Navier-Stokes equation is posed as a nonlinear matrix differential equation. When the flow is assumed to be a time series having unknown wavenumber coefficients, then the matrix in this periodic Navier-Stokes matrix differential equation becomes a time series of matrices. Posed in this way, any flow's unknown wavenumber coefficients can be solved for recursively beginning with the zeroth-order coefficient representing any initial flow. When all matrices in the time series commute, a flow's unknown coefficients also represent the Taylor expansion of a stable matrix exponential product operating on the initial flow. This paper argues that a solution's coefficients converge to these bounded Taylor coefficients when these bounds are evaluated with a general solution's noncommutative matrices. | On the convergence of periodic Navier-Stokes flows | 11,650 |
We consider the Derivative NLS equation with general quadratic nonlinearities. In \cite{be2} the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$. Here we prove a similar result for large initial data in all dimensions $n \geq 2$. | Large data local solutions for the derivative NLS equation | 11,651 |
We prove the existence of a weak solution to a two-dimensional resonant 3x3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compensated compactness argument adapted from the paper of Tadmor, Rascle and Bagnerini, "Compensated compactness for 2D conservation laws", [JHDEs 2(3):697--712, 2005]. Existence is proved under the assumption that the flux functions in the two directions are linearly independent. | On the existence and compactness of a two-dimensional resonant system of
conservation laws | 11,652 |
We construct the precise boundary trace of positive solutions of $\Delta u=u^q$ in a smooth bounded domain in $R^N$, for $q$ in the super-critical range $q\geq (N+1)/(N-1)$. The construction is performed in the framework of the fine topology associated with the Bessel capacity $C_{2/q,q'}$ on the boundary of the domain. We prove that the boundary trace is a Borel measure (in general unbounded), which is outer regular relative to this capacity. We provide a necessary and sufficient condition for such measures to be the boundary trace of a positive solution and prove that the corresponding generalized boundary value problem is well-posed in the class of $\sigma$-moderate solutions. | The precise boundary trace of positive solutions of the equation $Δ
u=u^q$ in the supercritical case | 11,653 |
For higher order integral functionals with $p(x)$ growth with respect to the highest order derivative $D^m u$, we prove that $D^m u$ is H\"older continuous on an open subset $\Omega_0 \subset \Omega$ of full Lebesgue- measure, provided that the exponent function $p:\Omega \to (1,\infty)$ itself is H\"older continuous. | Partial regularity for minima of higher order functionals with p(x)
growth | 11,654 |
For weak solutions $u \in W^{m,1}(\Omega;\R^N)$ of higher order systems of the type \int_\Omega < A(x,D^m u),D^m \phi > dx = \int_\Omega < |F|^{p(x)-2}F,D^m \phi> dx, for all $\phi \in C^{\infty}_c(\Omega;\R^N), m > 1$ with variable growth exponent $p:\Omega \to (1,\infty)$ we prove that if $|F|^{p(\cdot)} \in L^q_{loc}(\Omega)$ with $1 < q < \frac{n}{n-2} + \delta$, then $|D^m u|^{p(\cdot)} \in L^q_{loc}(\Omega)$. We should note that we prove this implication both in the non-degenerate and in the degenerate case. | Calderón-Zygmund estimates for higher order systems with p(x) growth | 11,655 |
We show that bounded families of global classical relativistic strings that can be written as graphs are relatively compact in C0 topology, but their accumulation points include many non relativistic strings. We also provide an alternative formulation of these relativistic strings and characterize their ``semi-relativistic'' completion. | Non relativistic strings may be approximated by relativistic strings | 11,656 |
We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely). | Global well-posedness for the KP-I equation on the background of a non
localized solution | 11,657 |
We study the dispersive properties of the Schr\"odinger equation. Precisely, we look for estimates which give a control of the local regularity and decay at infinity {\it separately}. The Banach spaces that allow such a treatment are the Wiener amalgam spaces, and Strichartz-type estimates are proved in this framework. These estimates improve some of the classical ones in the case of large time. | Strichartz Estimates in Wiener Amalgam Spaces for the Schrödinger
equation | 11,658 |
We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao. | Global well-posedness, scattering and blow-up for the energy-critical,
focusing, non-linear Schrodinger equation in the radial case | 11,659 |
In this article we study the existence, continuation and bifurcation from infinity of nonconstant solutions for a nonlinear Neumann problem. We apply the Leray-Schauder degree and the degree for SO(2)-equivariant gradient operators defined by the second author. | Existence and continuation of solutions for a nonlinear Neumann problem | 11,660 |
We are concerned with singular elliptic equations of the form $-\Delta u= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with sublinear growth and $g$ is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle. | Ground state solutions for the singular Lane-Emden-Fowler equation with
sublinear convection term | 11,661 |
If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $X\times X^*$, then the vector field $x\to \bar\partial L(x):=\{p\in X^*; (p,x)\in \partial L(x,p)\}$ is maximal monotone. Conversely, any maximal monotone operator $T$ on $X$ is derived from such a potential on phase space, that is there exists a selfdual Lagrangian $L$ on $X\times X^*$ (i.e, $L^*(p, x) =L(x, p)$) such that $T=\bar\partial L$. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form $T=\partial \phi$ for some convex lower semi-continuous function on $X$. This representation allows for the application of the selfdual variational theory --recently developed by the author-- to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form $\Lambda x\in Tx$ for a given map $\Lambda: D(\Lambda)\subset X\to X^*$, can now be obtained by minimizing functionals of the form $I(x)=L(x,\Lambda x)-< x, \Lambda x>$. | Maximal monotone operators are selfdual vector fields and vice-versa | 11,662 |
The multivariate analogue of Dalamber's equation in the space of generalized functions is considered. The method of generalized functions for the building of solutions of nonstationary boundary value problems for wave equations in spaces of different dimensions is elaborated. Dynamic analogues of Green and Gauss formulas for solutions of wave equation in the space of generalized functions are built. Their regular integral representations and singular boundary integral equations for solving the nonstationary problems are constructed for the spaces of the dimensions 1,2,3. The method of obtaining of conditions on fronts of shock waves is stated. | Nonstationary boundary value problems for wave equation and their
generalized solutions | 11,663 |
The purpose of this paper is to set out optimal gradient estimates for solutions to the isotropic conductivity problem in the presence of adjacent conductivity inclusions as the distance between the inclusions goes to zero and their conductivities degenerate. This difficult question arises in the study of composite media. Frequently in composites, the inclusions are very closely spaced and may even touch. It is quite important from a practical point of view to know whether the electric field (the gradient of the potential) can be arbitrarily large as the inclusions get closer to each other or to the boundary of the background medium. In this paper, we establish both upper and lower bounds on the electric field in the case where two circular conductivity inclusions are very close but not touching. We also obtain such bounds when a circular inclusion is very close to the boundary of a circular domain which contains the inclusion. The novelty of these estimates, which improve and make complete our earlier results published in Math. Ann., is that they give an optimal information about the blow-up of the electric field as the conductivities of the inclusions degenerate. | Optimal Estimates for the Electric Field in Two-Dimensions | 11,664 |
Let $\Omega\subset\mathbb{R}^{N}$, $N\geq2$ be a bounded smooth domain and $\alpha>1$. We are interested in the singular elliptic equation% \[ \triangle h=\frac{1}{\alpha}h^{-\alpha}-p\quad\text{in}\Omega \] with Neumann boundary conditions. In this paper, we gave a complete description of all continuous radially symmetric solutions. In particular, we constructed nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N=2 and $\alpha=3$, the equation has been used to model steady states of van der Waals force driven thin films of viscous fluids. We also considered the physical problem when total volume of the fluid is prescribed. | On steady states of van der Waals force driven thin film equations | 11,665 |
We consider positive solutions of the stationary Gierer-Meinhardt system. Under suitable conditions on the exponents $p,q,r$ and $s$, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both positive and zero source production cases. | A priori estimates of stationary solutions of an activator-inhibitor
system | 11,666 |
We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the L^p norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data. | The regularity and Neumann problem for non-symmetric elliptic operators | 11,667 |
A class of diffusion driven Free Boundary Problems is considered which is characterized by the initial onset of a phase and by an explicit kinematic condition for the evolution of the free boundary. By a domain fixing change of variables it naturally leads to coupled systems comprised of a singular parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even though the one dimensional case has been thoroughly investigated, results as basic as well-posedness and regularity have so far not been obtained for its higher dimensional counterpart. In this paper a recently developed regularity theory for abstract singular parabolic Cauchy problems is utilized to obtain the first well-posedness results for the Free Boundary Problems under consideration. The derivation of elliptic regularity results for the underlying static singular problems will play an important role. | A Class of Free Boundary Problems with Onset of a new Phase | 11,668 |
Let G be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy be the multiplication operator. We show the associated fibration m mapping GxG to G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show the pull back of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions that the pull back of a form on the base is harmonic on the total space. | The spectral geometry of the canonical Riemannian submersion of a
compact Lie Group | 11,669 |
A general class of singular abstract Cauchy problems is considered which naturally arises in applications to certain Free Boundary Problems. Existence of an associated evolution operator characterizing its solutions is established and is subsequently used to derive optimal regularity results. The latter are well known to be important basic tools needed to deal with corresponding nonlinear Cauchy Problems such as those associated to Free Boundary Problems. | Optimal Regularity for a Class of Singular Abstract Parabolic Equations | 11,670 |
We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy holds: a) if the homogeneous Sobolev norm H^1 of u_0 is smaller than that of W, we have global well-posedness and scattering, b) if the homogeneous Sobolev norm H^1 of u_0 is larger than that of W, there is blow-up in finite time. Our general approach is the one we introduced in our previous work on the corresponding problem for NLS (math.AP/0610266, Inventiones Math 2006, Online First), where we proved the corresponding result for NLS in the radial case. In the case of the wave equation we are able to treat general data by using a further conservation law in the energy space, the finite speed of propagation and Lorentz transformations to establish a crucial orthogonality property for " energy critical " elements. To prove the required rigidity theorem in the case of blow-up in finite time, we cannot use the invariance of the L^2 norm as in the case of NLS. Instead, (following earlier work of Merle-Zaag and of Giga-Kohn in the parabolic case) we introduce self-similar variables. We thus find a further Liapunov function which allows us to reduce matters to a degenerate elliptic problem with critical non-linearity. We use unique continuation to rule out the existence of non-zero solutions for the degenerate elliptic problem, thus completing the proof. | Global well-posedness, scattering and blow-up for the energy critical
focusing non-linear wave equation | 11,671 |
An $L_{q}(L_{p})$-theory of divergence and non-divergence form parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all measurable functions depending only on $t$. The method of proving simplifies the methods previously used in the case $p=q$. | Parabolic equations with VMO coefficients in spaces with mixed norms | 11,672 |
In this paper, we establish rigorous existence theorems for a mathematical model of a thin inflated wrinkled membrane that is subjected to a shape dependent hydrostatic pressure load. We are motivated by the problem of determining the equilibrium shape of a strained high altitude large scientific balloon. This problem has a number of unique features. The balloon is very thin (30 micron), especially when compared with its diameter (over 100 meters). Unlike a standard membrane, the balloon is unable to support compressive stresses and will wrinkle or form folds of excess material. Our approach can be adapted to a wide variety of inflatable membranes, but we will focus on two types of high altitude balloons, a zero-pressure natural shape balloon and a super-pressure pumpkin shaped balloon. We outline the shape finding process for these two classes of balloon designs, formulate the problem of a strained balloon in an appropriate Sobolev space setting, establish rigorous existence theorems using direct methods in the calculus of variations, and present numerical studies to complement our theoretical results. | Existence theorems for thin inflated wrinkled membranes subjected to a
hydrostatic pressure | 11,673 |
In a previous work, we presented a class of initial data to the three dimensional, periodic, incompressible Navier-Stokes equations, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch-Tataru type space, then there is a global solution to the Navier-Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large in $ C^{-1}$. Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction by the authors and H. Bahouri, thus generating a large number of different examples. | Wellposedness and stability results for the Navier-Stokes equations in
${\mathbf R}^{3}$ | 11,674 |
We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We formulate such question as an inverse problem for an integro-differential equation posed on the half line. We develop firstly a regular dependency theory for the solution in terms of the coefficients and, secondly, a novel regularization technique for tackling this inverse problem which takes into account the specific nature of the equation. Our results rely also on generalized relative entropy estimates and related Poincar\'e inequalities. | On the Inverse Problem for a Size-Structured Population Model | 11,675 |
We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's paraproduct decomposition, it is proved that strong solution $(u,b)$ can be extended after $t=T$ if either $u\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 1$ and $b\in L^1_T(\dot B^{0}_{\infty,\infty})$, or $(\omega, J)\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 2$, where $\omega(t)=\na\times u $ denotes the vorticity of the velocity and $J=\na\times b$ the current density. | Existence theorem and blow-up criterion of strong solutions to the
two-fluid MHD equation in ${\mathbb R}^3$ | 11,676 |
We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for a dispersion generalised KP II type equation. We also deduce a global well-posedness result for the generalised equation. | Well-posedness for the Kadomtsev-Petviashvili II equation and
generalisations | 11,677 |
Consider the mean field equation with critical parameter $8\pi$ in a bounded smooth domain $\Omega$. Denote by $E_{8\pi}(\Omega)$ the infimum of the associated functional $I_{8\pi}(\Omega)$. We call $E_{8\pi}(\Omega)$ the "energy" of the domain $\Omega$. We prove that if the area of $\Omega$ is equal to $\pi$, then the energy of $\Omega$ is always greater or equal to the energy of the unit disk and equality holds if and only if $\Omega$ is the unit disk. We also give a sufficient condition for the existence of a minimizer for $I_{8\pi}(\Omega)$. | The Mean Field Equation with Critical Parameter in a Plane Domain | 11,678 |
In this note, we consider wave packet parametrices for Schrodinger-like evolution equations. Under an integrability condition along the flow, we prove that the flow is then globally well-defined and bilipschitz. Under an additional smallness assumption, we prove that the kernel of the phase space operator decays rapidly away from the graph of the Hamilton flow. This can be used to prove that smoothness is preserved for finite times by the flow of the equation. | Wave packet parametrices for evolutions governed by PDO's with rough
symbols | 11,679 |
We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four sharp inequalities, which state the existences of the corresponding extremal metrics. | Steady States for One Dimensional Conformal Metric Flows | 11,680 |
We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform. | Propagation of Gevrey regularity for solutions of the Boltzmann equation
for Maxwellian molecules | 11,681 |
In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial value is smooth and periodic, and $\left\| \nabla \theta_0 \right\|_{L^{\infty}}^{1 - 2 s} \left\| \theta_0 \right\|_{L^{\infty}}^{2 s}$ is small, where $s$ is the power of the fractional Laplacian, then no finite time singularity will occur for the super-critically dissipative 2D quasi-geostrophic equation. | Remarks on the Global Regularity for the Super-Critical 2D Dissipative
Quasi-Geostrophic Equation | 11,682 |
Let $X$ be a compact K\"ahler manifold. The set $\cha(X)$ of one-dimensional complex valued characters of the fundamental group of $X$ forms an algebraic group. Consider the subset of $\cha(X)$ consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree $d$. This set is shown to be a union of finitely many components that are translates of algebraic subgroups of $\cha(X)$. When the degree $d$ equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is shown that the number of equivalence classes (under a natural equivalence relation) of holomorphic maps, with connected fibers, of $X$ onto smooth curves of a fixed genus $>1$ is a topological invariant of $X$. In fact it depends only on the fundamental group of $X$. | Higgs line bundles, Green-Lazarsfeld sets,and maps of Kähler manifolds
to curves | 11,683 |
We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a ``cohomology theory" for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to $L$-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational $(p,p)$-cohomology class. | A theory of algebraic cocycles | 11,684 |
The space of holomorphic maps from $S^2$ to a complex algebraic variety $X$, i.e. the space of parametrized rational curves on $X$, arises in several areas of geometry. It is a well known problem to determine an integer $n(D)$ such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension $n(D)$, where $D$ denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and $n(D)\to\infty$ as $D\to\infty$. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, $n(D)$ may be computed, and $n(D)\to\infty$ as $D\to\infty$. For other singular toric varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily large by a suitable choice of $D$. | Configuration spaces and the space of rational curves on a toric variety | 11,685 |
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in affine space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactification instead of always resorting to products of projective spaces. | Toric Intersection Theory for Affine Root Counting | 11,686 |
For any Lagrangean K\"ahler submanifold $M \subset T^*{\Bbb C}^n$, there exists a canonical hyper K\"ahler metric on $T^*M$. A K\"ahler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper K\"ahler manifolds with large automorphism group. Using it, a class of pseudo hyper K\"ahler manifolds of complex signature $(2,2n)$ is constructed. For any hyper K\"ahler manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper K\"ahler metric of complex signature $(2,2n)$. | On Hyper Kähler manifolds associated to Lagrangean Kähler
submanifolds of $T^*{\Bbb C}^n$ | 11,687 |
The purpose of this paper is to translate positivity properties of the tangent bundle (and the anti-canonical bundle) of an algebraic manifold into existence and movability properties of rational curves and to investigate the impact on the global geometry of the manifold $X$. Among the results we prove are these: \quad If $X$ is a projective manifold, and ${\cal E} \subset T_X$ is an ample locally free sheaf with $n-2\ge rk {\cal E}\ge n$, then $X \simeq \EP_n$. \quad Let $X$ be a projective manifold. If $X$ is rationally connected, then there exists a free $T_X$-ample family of (rational) curves. If $X$ admits a free $T_X$-ample family of curves, then $X$ is rationally generated. | Rational curves and ampleness properties of the tangent bundle of
algebraic varieties | 11,688 |
In this paper, we attempt to determine the quantum cohomology of projective bundles over the projective space P^n. In contrast to the previous examples, the relevant moduli spaces in our case frequently do not have expected dimensions. It makes the calculation more difficult. We overcome this difficulty by using excessive intersection theory. | Quantum cohomology of projective bundles over P^n | 11,689 |
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2. | Extensions of vector bundles and rationality of certain moduli spaces of
stable bundles | 11,690 |
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first part of the 16th Hilbert problem appear in this classification. One is the distribution of the components of the curve between the components of the non-connected cubic surface which turns out to depend on the patterns of arrangements (see Theorem 1). The other is presence of positive genus among the components of the complement and genus-related restrictions (see Theorems 3 and 4). | Topological arrangement of curves of degree 6 on cubic surfaces in $\Bbb
R P^3$ | 11,691 |
The zero set of a real polynomial in two variable is a curve in $\mathbb R^2$. For a generic choice of its coefficients this is a non-singular curve, a collection of circles and lines properly embedded in $\mathbb R^2$. What topological arrangements of these circles and lines appear for the polynomials of a given degree? This question arised in the 19th century in the works of Harnack and Hilbert and was included by Hilbert into his 16th problem. Several partial results were obtained since then. However the complete answer is known only for polynomials of degree 5 or less. The paper presents a new partial result toward the solution of the 16th Hilbert problem. The proof makes use of the proof by Kronheimer and Mrowka of the Thom conjecture in $\mathbb C P^2$. | Adjunction inequality for real algebraic curves | 11,692 |
The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\ell$-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases. | Subspace Arrangements over Finite Fields: Cohomological and Enumerative
Properties | 11,693 |
We study the rigidity questions and the Albanese Variety for Complex Parallelizable Manifolds. Both are related to the study of the cohomology group $H^1(X,\mathcal O)$. In particular we show that a compact complex parallelizable manifold is rigid iff $b_1(X)=0$ iff Alb$(X)=\{e\}$ iff $H^1(X,\mathcal O)=0$. | On Rigidity and the Albanese Variety for Parallelizable Manifolds | 11,694 |
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an exact description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are immediately obvious. | Pieri-type formulas for maximal isotropic Grassmannains via triple
intersections | 11,695 |
Looijenga recently proved that the tautological ring of M_g vanishes in degree d>g-2 and is at most one-dimensional in degree g-2, generated by the class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g. The proof uses the Witten conjecture, proven by Kontsevich. With similar methods, we expect to be able to prove some (possibly all) of the identities in degree g-2 in the tautological ring that are part of the author's conjectural explicit description of the ring. | A non-vanishing result for the tautological ring of {\cal M}_g | 11,696 |
Let $V$ be a smooth, projective, convex variety. We define tautological $\psi$ and $\kappa$ classes on the moduli space of stable maps $\M_{0,n}(V)$, give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of $V$ twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of $V$. This results in families of Frobenius manifold structures on the cohomology ring of $V$ which includes the quantum cohomology as a special case. | Intersection Numbers on the Moduli Spaces of Stable Maps in Genus 0 | 11,697 |
We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding \'etale cohomology sheaves do fulfill Bloch-Ogus theory. | Germs of de Rham cohomology classes which vanish at the generic point | 11,698 |
We study tori attached to the fundamental groups of plane curves with arbitrary singularities. These tori provide complete information about homology of finite abelian covers of the plane branched along the curve. We calculate these tori in terms of certain linear systems determined by the singularities of the curve. In the case of the complements to a union of lines they can be calculated from the lattice of the arrangement and are closely related to the components of the space of Aomoto complexes with prescribed homology. | Characteristic varieties of algebraic curves | 11,699 |
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