Showing papers in "Communications on Pure and Applied Mathematics in 2004"
TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.
4,339 citations
TL;DR: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C2 edges.
Abstract: This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j/2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior.
1,567 citations
TL;DR: In this article, the authors considered the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom and constructed two-dimensional inviscid periodic traveling waves with vorticity.
Abstract: We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. (C) 2004 Wiley Periodicals, Inc.
480 citations
TL;DR: In this article, the authors present a method to prove convergence of gradient flows of families of energies that Γ-converge to a limiting energy and provide lower-bound criteria to obtain the convergence that correspond to a sort of C 1-order Γ -convergence of functionals.
Abstract: We present a method to prove convergence of gradient flows of families of energies that Γ-converge to a limiting energy. It provides lower-bound criteria to obtain the convergence that correspond to a sort of C1-order Γ-convergence of functionals. We then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy in dimension 2, retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field. © 2004 Wiley Periodicals, Inc.
334 citations
TL;DR: In this paper, the least-action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise.
Abstract: The least-action principle from the Wentzell-Freidlin theory of large deviations is exploited as a numerical tool for finding the optimal dynamical paths in spatially extended systems driven by a small noise. The action is discretized and a preconditioned BFGS method is used to optimize the discrete action. Applications are presented for thermally activated reversal in the Ginzburg-Landau model in one and two dimensions, and for noise-induced excursion events in the Brusselator taken as an example of a nongradient system arising in chemistry. In the Ginzburg-Landau model, the reversal proceeds via interesting nucleation events, followed by propagation of domain walls. The issue of nucleation versus propagation is discussed, and the scaling for the number of nucleation events as a function of the reversal time and other material parameters is computed. Good agreement is found with the numerical results. In the Brusselator, whose deterministic dynamics has a single stable equilibrium state, the presence of noise is shown to induce large excursions by which the system cycles out of this equilibrium state. c � 2004 Wiley Periodicals, Inc.
272 citations
TL;DR: In this paper, the optimal regularity requirements on the coefficients and domains for the W1,p theory, 1 < p < ∞, were obtained, where the principal coefficients are in the John-Nirenberg space with small BMO seminorms.
Abstract: The inhomogeneous Dirichlet problems concerning divergence form elliptic equations are studied. Optimal regularity requirements on the coefficients and domains for the W1,p theory, 1 < p < ∞, are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO seminorms. The domain is a Reifenberg domain. These conditions for the W1,p theory not only weaken the requirements on the coefficients but also lead to a more general geometric condition on the domains. In fact, these domains might have fractal dimensions. © 2004 Wiley Periodicals, Inc.
263 citations
TL;DR: In this article, the authors proved global existence and scattering for the defocusing, cubic nonlinear Schrodinger equation in H s (R 3 ) for s > 4. But they did not consider the case where s > 2 and the main new estimate in the argument is a Morawetz-type inequality for the solution.
Abstract: We prove global existence and scattering for the defocusing, cubic nonlinear Schrodinger equation in H s (R 3 ) for s > 4 . The main new estimate in the argument is a Morawetz-type inequality for the solution �. This estimate bounds k �(x,t)k L 4(R3×R) , whereas the well-known Morawetz-type estimate of Lin-Strauss controls R 1 0 R R3 (�(x,t))4 |x| dxdt.
254 citations
TL;DR: In this article, the authors derive several asymptotic formulae that are of practical interest, for example, in the calibration problem, which is an inverse problem that consists
Abstract: The Black-Scholes model [6, 23] has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models, the volatility is assumed to depend on the variables t (time) and S, giving rise to the so-called local volatility models. The second one, conceptually more ambitious, considers that the volatility has a stochastic component of its own. In the latter, the number of factors is increased by the amount of stochastic factors entering the volatility modeling. Both models are of practical interest. In these contexts, it is relevant to express the resulting prices in terms of implied volatilities. Given a price, the Black-Scholes implied volatility is determined, for each given product (that is for each given strike and expiry date defining, say, the call option) as the unique value of the volatility parameter for which the BlackScholes pricing formula agrees with that given price. Actually, it is common practice on trading floors to quote and to observe prices in this way. A great advantage of having prices expressed in such dimensionless units is to provide easy comparison between products with different characteristics. In principle, the implied volatility can be inferred from computed options prices by inverting the Black-Scholes formula. It is more convenient, however, to directly analyze the implied volatility. Indeed, this approach allows us to shed light on qualitative properties that would otherwise be more difficult to establish. In particular, we derive here several asymptotic formulae that are of practical interest, for example, in the calibration problem. The latter—an inverse problem that consists
249 citations
TL;DR: The total‐variation‐based image denoising model of Rudin, Osher, and Fatemi can be generalized in a natural way to favor certain edge directions and the resulting anisotropic energies and study properties of their minimizers are considered.
Abstract: The total-variation-based image denoising model of Rudin, Osher, and Fatemi can be generalized in a natural way to favor certain edge directions. We consider the resulting anisotropic energies and study properties of their minimizers. © 2004 Wiley Periodicals, Inc.
219 citations
TL;DR: The pseudo-spectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics as mentioned in this paper, and it affects the study of the stability of laminar flows.
Abstract: The pseudo-spectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. For example, in computational fluid dynamics it affects the study of the stability of laminar flows. In fact, even for the most basic flows, the computations entirely fails to predict what is observed in the experiments. The explanation is that for non-normal operators the resolvent could be very large far away from the spectrum, which makes computation of the eigenvalues impossible. The occurence of ``false eigenvalues'' is due to the existence of quasi-modes, i.e., approximate local solutions to the eigenvalue problem. The quasi-modes appear since the Nirenberg-Treves condition (Psi) is not satisfied for topological reasons. (Less)
176 citations
TL;DR: In this paper, the authors studied radially symmetric solutions of the nonlinear heat equation and derived various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup.
Abstract: In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = 1 u + |u| p−1 u either on R N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, p > ps := N + 2 N − 2 . We prove that if ps p ∗ , the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. W e then establish useful estimates for the so-called incomplete blowup, which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. c 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the authors considered the self-similarity in Smoluchowski's equations of coagulation for the solvable kernels K (x, y) = 2, x + y, and xy.
Abstract: We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K (x, y) = 2, x + y, and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay whose form is related to heavy-tailed distributions well-known in probability theory. For K = 2 the size distribution is Mittag-Leffler, and for K = x + y and K = xyit is a power-law rescaling of a maximally skewed α-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits. c � 2004 Wiley Periodicals, Inc.
TL;DR: For the stable, forward Muskat problem, this paper showed global-in-time existence for initial data that is a small perturbation of a flat interface, and for the unstable, backwa rd problem, in which the higher-viscosity fluid contracts, they constructed singular solution s that start off with smooth initial data but develop a point of infinite curvature at finite time.
Abstract: The Muskat, or Muskat-Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contras t to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem . For the stable, forward Muskat problem, in which the higher-viscosity fluid ex pands into the lower-viscosity fluid, we show global-in-time existence for initial data that is a small perturbation of a flat interface. The initial data in this result ma y contain weak (e.g., curvature) singularities. For the unstable, backwa rd problem, in which the higher-viscosity fluid contracts, we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time. c 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors studied pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold and proved that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension.
Abstract: We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension.
TL;DR: In this article, the leading-order term of the solution of the focusing nonlinear (cubic) Schrodinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions was derived.
Abstract: We calculate the leading-order term of the solution of the focusing nonlinear (cubic) Schrodinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. We utilize the Riemann-Hilbert problem formulation of the inverse scattering problem to obtain the leading-order term of the solution. Error estimates are provided. © 2004 Wiley Periodicals, Inc.
TL;DR: The Camassa-Holm equation as discussed by the authors shows that the flow may break down at some point when the auxiliary function m = v - v'' is a caricature of the vorticity that controls the behavior of honest Euler in dimensions 2 and 3; see Bertozzi and Majda for a full account.
Abstract: The Camassa-Holm equation is written in its Eulerian form as
$$\displaystyle\begin{array}{rcl} \text{CH:}\quad \frac{\partial v} {\partial t} + v\frac{\partial v} {\partial x} + \frac{\partial p} {\partial x} = 0,& & {}\\ \end{array}$$
with “pressure” \(p = G[v^{2} + \frac{1} {2}(v')^{2}]\). Here \(G = (1 - D^{2})^{-1} = \frac{1} {2}e^{-\vert x-y\vert }\). It is easy to see that if, at time t = 0, v is odd with v(x) > 0 for x < 0 and v′(0) < 0, then the slope s(t) = v′(t, 0) satisfies \(s^{\bullet } < -\frac{1} {2}s^{2}\) and so is driven down to \(-\infty \) at some time \(T \leq -2/s(0)\): in short, the flow may “break down.” This is the “steepening lemma” of Camassa and Holm [2]. Nothing worse happens: v(t, x) itself cannot jump like the usual kind of shock. Actually, it is not the obvious v but rather the auxiliary function \(m = v - v''\) that controls this; m is a caricature of the vorticity that controls the behavior of honest Euler in dimensions 2 and 3; see Bertozzi and Majda [1] for a full account.
TL;DR: In this article, the authors studied the pointwise behavior of the Green's function of the Boltzmann equation and showed that the particle and fluid aspects of the equation are represented by singular waves.
Abstract: We study the pointwise behavior of the Green’s function of the Boltzmann equation. Our results reveal the particle and fluid aspects of the equation. The particle aspect is represented by singular waves. These waves are carried by transport equations and dominate the short-time behavior of the solution. We devise a Picard-type iteration for constructing the increasingly regular particlelike waves. The fluidlike waves reveal the dissipative behavior of the type of Navier-Stokes equations as usually seen by the Chapman-Enskog expansion. These waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable. The fluidlike waves represent the long-time behavior of the solution. As an application, we obtain the pointwise description of the large-time behavior of the convergence to the global Maxwellian when the initial perturbation is not necessarily smooth. In our analysis of the exchanges of the microscopic velocity decay and space decay, we make essential use of the hard sphere models. c 2004 Wiley Periodicals, Inc. Contents
TL;DR: In this paper, it was shown that a stationary extrinsic (or intrinsic) biharmonic map u ∈ W2,2(Ω, N) from Ω ⊂ Rm into a Riemnanian manifold N is smooth away from a closed set of (m − 4)-dimensional Hausdorff measure zero.
Abstract: For m ≥ 5, we prove that a stationary extrinsic (or intrinsic, respectively) biharmonic map u ∈ W2,2(Ω, N) from Ω ⊂ Rm into a Riemnanian manifold N is smooth away from a closed set of (m − 4)-dimensional Hausdorff measure zero. © 2003 Wiley Periodicals, Inc.
TL;DR: In this paper, the existence and stability of multidimensional transonic shocks (hyperbolic-elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders was established.
Abstract: We establish the existence and stability of multidimensional transonic shocks (hyperbolic-elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C1,α, provided that the hyperbolic phase is close in C1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C2,α, the free boundary is C2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.
TL;DR: In this paper, the authors revisited the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linear unstable equilibrium, and gave sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable.
Abstract: We revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linearly unstable equilibrium. For a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension. We introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism. We obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable. Our method also allows us to obtain efficient lower and upper bounds for the propagation speed. c
TL;DR: In this paper, the authors prove the global existence and convergence of the mean curvature flow of the graph of a smooth map between unit spheres of possibly different dimensions under various conditions.
Abstract: Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is isotopic to a constant map. c � 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, a nonlinear orbital stability of small-amplitude Lax-type viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [20] was established.
Abstract: Combining pointwise Green's function bounds obtained in a companion paper [36] with earlier, spectral stability results obtained in [16], we establish nonlinear orbital stability of small-amplitude Lax-type viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [20], notably including compressible Navier-Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in Lp with respect to small L1 ∩ H3 perturbations, 2 ≤ p ≤ ∞. Our analysis extends and somewhat refines the approach introduced in [35] to treat stability of relaxation profiles. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the infinite Prandtl number model of convection is shown to be the limit of the Boussinesq approximation to the Rayleigh-Benard convection as the number of layers approaches infinity.
Abstract: We rigorously justify the infinite Prandtl number model of convection as the limit of the Boussinesq approximation to the Rayleigh-Benard convection as the Prandtl number approaches infinity. This is a singular limit problem involving an initial layer. © 2003 Wiley Periodicals, Inc.
TL;DR: In this paper, a three-dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations is presented, which has structural similarities with the Euler equations for incompressible, inviscid fluid flows.
Abstract: We present a three-dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler equations for incompressible, inviscid fluid flows. It mimics certain important properties of the Euler equations, namely, conservation of energy and divergence-free velocity. It is proven for certain families of initial data that the model system permits local existence in time for initial conditions in Sobolev spaces Hs, s > ; and blowup occurs in the sense that the H3/2 + ϵ norm becomes unbounded in finite time. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, sharp constants for the Moser-Trudinger inequalities on complex spheres in complex space C n were derived for smooth functions and holomorphic functions with different sharp constants.
Abstract: The main results of this paper concern sharp constants for the Moser-Trudinger inequalities on spheres in complex space C n . We derive Moser-Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser-Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser-Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser-Onofri-type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4). c 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's "low-curvature image simplifiers" for image denoising and segmentation is considered.
Abstract: We consider a class of fourth-order nonlinear diffusion equations motivated by Tumblin and Turk's “low-curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form
where g(s) = k2/(k2 + s2) is a “curvature” threshold and λ denotes a fidelity-matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite-time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second-order Perona-Malik equation (an ill-posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth-order lubrication-type equations and the design of positivity-preserving schemes for such equations. This connection also has relevance for other related fourth-order imaging equations. © 2004 Wiley Periodicals, Inc.
TL;DR: In this article, the authors consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation and analyze the asymptotics of solutions to these equations.
Abstract: We consider a class of space-times for which the essential part of Einstein’s equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1+1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations ast →∞ . For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t −1/2 as t →∞ . In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): (1) The solution converges to a point. (2) The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). (3) The solution goes to infinity along a curve y = const. (4) The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. c � 2004 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors considered locally convex hypersurfaces, where the function f is defined in the convex cone Γn ≡ { λ ∈ R : each component λi > 0 } and satisfy the fundamental structure conditions.
Abstract: for some constant K, where κ[M ] = (κ1, . . . , κn) denotes the principal curvatures of M . Important examples include the classical Plateau problem for minimal or constant mean curvature surfaces and the corresponding problem for Gauss curvature, which was treated recently by the authors [12] and independently by Trudinger-Wang [24]. In this paper as in our previous work [12], we are concerned with locally convex hypersurfaces. Accordingly, the function f is assumed to be defined in the convex cone Γn ≡ { λ ∈ R : each component λi > 0 } in R and satisfy the fundamental structure conditions:
TL;DR: In this article, the authors propose a method to solve the problem of the problem: this article...,.. ].. ).. ]... )...
Abstract: CONTENTS
TL;DR: In this article, it was shown that if the Fermi level lies in a spectral gap, then in the adiabatic limit the accumulated excess Hall charge is given by the linear response Kubo-Streda formula.
Abstract: The adiabatic charge transport is investigated in a two-dimensional Landau model perturbed by a bounded potential at zero temperature. We show that if the Fermi level lies in a spectral gap, then in the adiabatic limit the accumulated excess Hall charge is given by the linear response Kubo-Streda formula. The proof relies on the expansion of Nenciu, some generalized phase space estimates, and a bound on the speed of propagation. © 2004 Wiley Periodicals, Inc.