Showing papers in "Communications on Pure and Applied Mathematics in 2003"
TL;DR: In this paper, the degree-counting formula for (0.1) is given for ρ ∈ (8mπ, 8m + 1)π, where ρ is a real number.
Abstract: We consider the following mean field equations:
(0.1)
where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number,
(0.2)
and where Ω is a bounded smooth domain, h is a C1 positive function on Ω, and ρ ∈ ℝ. Based on our previous analytic work [14], we prove, among other things, that the degree-counting formula for (0.1) is given by () for ρ ∈ (8mπ, 8(m + 1)π). © 2003 Wiley Periodicals, Inc.
309 citations
300 citations
TL;DR: In this paper, Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem are discussed. But these results are not applicable to the Niren-Berg problem.
Abstract: We will report some results concerning the Yaniabe problem and the Niren berg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem.
251 citations
TL;DR: In this paper, the authors investigated the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by Francfort and Marigo in [15] and showed that the notion of minimizer of a Mumford and Shah type function for its own jump set is stable under weak convergence assumptions.
Abstract: This paper investigates the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by Francfort and Marigo in [15]. The starting point is a time discretized version of that evolution which results in a sequence of minimization problems of Mumford and Shah type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of the time-continuous quasi-static growth is to pass to the limit as the time-discretization step tends to 0. This is performed with the help of a jump transfer theorem which permits, under weak convergence assumptions for a sequence {un} of SBV-functions to its BV-limit u, to transfer the part of the jump set of any test field that lies in the jump set of u onto that of the converging sequence {un}. In particular, it is shown that the notion of minimizer of a Mumford and Shah type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time-continuous quasi-static evolution. © 2003 Wiley Periodicals, Inc.
211 citations
TL;DR: In this article, the existence of the double scaling limit in the unitary matrix model with quartic interaction was proved based on the Riemann-Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points.
Abstract: We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the ψ function for the Hastings-McLeod solution to the Painleve II equation. The proof is based on the Riemann-Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points. © 2003 Wiley Periodicals, Inc.
201 citations
TL;DR: In this article, the authors compute the long-time asymptotics for solutions of the NLS equation under the assumption that the initial data lies in a weighted Sobolev space.
Abstract: The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations.
200 citations
TL;DR: Singularities of wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into ℕ.
Abstract: Singularities of corotational wave maps from (1 + 2)-dimensional Minkowski space into a surface N of revolution after a suitable rescaling give rise to nonconstant corotational harmonic maps from 2 into ℕ. In consequence, for noncompact target surfaces of revolution, the Cauchy problem for wave maps is globally well-posed. © 2003 Wiley Periodicals, Inc.
164 citations
154 citations
TL;DR: In this paper, it was shown that G satisfies the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion-free elementary amenable or residually free, which implies that there are no nontrivial zero divisors in G. The result relies on new approximation results for L2-Betti numbers over G, which are the core of the work done in this paper.
Abstract: Let G be a torsion-free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion-free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2-Betti numbers over ℚG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number-theoretic properties of eigenvalues for the combinatorial Laplacian on L2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class . We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class . © 2003 Wiley Periodicals, Inc.
153 citations
TL;DR: In this paper, it was shown that the renormalized solutions of the Boltzmann equation considered in a bounded domain with different types of (kinetic) boundary conditions converge to the Stokes-Fourier system when the main free path goes to zero.
Abstract: We prove that the renormalized solutions of the Boltzmann equation considered in a bounded domain with different types of (kinetic) boundary conditions converge to the Stokes-Fourier system with different types of (fluid) boundary conditions when the main free path goes to zero. This extends the work of F. Golse and D. Levermore [9] to the case of a bounded domain. © 2003 Wiley Periodicals, Inc.
146 citations
TL;DR: In this article, the authors introduce the convex scattering support of a far field, a set which will be a subset of the hull of the support of any source which can produce it.
Abstract: We discuss inverse problems for the Helmholtz equation at fixed energy, specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle. We introduce the convex scattering support of a far field, a set which will be a subset of the convex hull of the support of any source which can produce it. We give several theorems which explain how to compute the convex scattering support and how to relate it to the actual support of a source, medium, or obstacle. © 2003 Wiley Periodicals, Inc.
TL;DR: In this article, the authors give an extension to the theorem of Jorgens, Calabi, and Pogorelov, showing that the only open convex subset of Rn which admits a convex C2 solution of det(D2u) = 1 in with limx→∂ u(x) = ∞ is Rn.
Abstract: (1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analytical proof, along the lines of affine geometry, of the theorem was later given by Cheng and Yau [9]. The first author extended the result for classical solutions to viscosity solutions [4]. It was proven by Trudinger and Wang in [19] that the only open convex subset of Rn which admits a convex C2 solution of det(D2u) = 1 in with limx→∂ u(x) = ∞ is = Rn . In this paper we give the following extension to the theorem of Jorgens, Calabi, and Pogorelov: Let u be a convex viscosity solution of det(D2u) = 1 outside a bounded subset of
TL;DR: In this article, the authors considered a three-dimensional planetary geostrophic viscous model of the gyre-scale mid-latitude ocean and established the existence of a nite-dimensional global attractor to this dissipative evolution system.
Abstract: In this paper we consider a three-dimensional planetary geostrophic viscous model of the gyre-scale mid-latitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a nite-dimensional global attractor to this dissipative evolution system. c 2003 Wiley Periodicals, Inc.
TL;DR: In this paper, a complete Riemannian manifold M is defined as a complete, non-compact manifold, and the Riesz transforms ∆ ∂ ∂xi 1−1/2, i = 1,..., n in Rn, and duality is established.
Abstract: Let M be a complete, noncompact Riemannian manifold, μ the Riemannian measure, ∇ the Riemannian gradient, and 1 the (positive) Laplace-Beltrami operator on M . Denote by | · | the length in the tangent space, and by ‖ · ‖p the norm in L p(M, μ), 1 ≤ p ≤ +∞. If one wants to define homogeneous Sobolev spaces of order one on M , that is, spaces of functions with one derivative in L p(M, μ), 1 < p < +∞, there are two obvious candidates for the seminorm: ‖|∇ f |‖p and ‖11/2 f ‖p. The former is local and of geometric nature, the latter is nonlocal and more analytic. When M is the Euclidean space, these two seminorms are equivalent for all p ∈ ]1, +∞[: C−1 p ‖1 f ‖p ≤ ‖|∇ f |‖p ≤ Cp‖1 f ‖p ∀ f ∈ C∞ 0 (R) . This relies on singular integral theory (see [40, 42]); indeed, the second inequality above is nothing but the L p-boundedness of the so-called Riesz transforms ∂ ∂xi 1−1/2, i = 1, . . . , n in Rn , and the first one follows by duality. It is, of course, a basic issue (which was raised in [43]) to ask for which complete, noncompact Riemannian manifolds M and which p ∈ ]1, +∞[ one has (1.1) C−1 p ‖1 f ‖p ≤ ‖|∇ f |‖p ≤ Cp‖1 f ‖p ∀ f ∈ C∞ 0 (M). For p = 2, on any complete Riemannian manifold, one has the equality
TL;DR: In this paper, the authors prove large deviation principles for random walks in a random environment: one for quenched walks in general ergodic environments and the other for averaged walks in product environments.
Abstract: We prove two large deviation principles for random walks in a random environment: one for quenched walks in general ergodic environments and the other for averaged walks in a product environment. © 2003 Wiley Periodicals, Inc.
TL;DR: In this paper, it was shown that the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system, in spite of the absence of linear instability.
Abstract: We consider Hamiltonian systems with U.1/ symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the absence of linear instability. Essentially, the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system. We apply our theory to the non
TL;DR: An infinite family of one‐bit sigma‐delta quantization schemes is constructed and it is proved that the error signal for π‐bandlimited signals is at most O(2−.07λ).
Abstract: One-bit quantization is a method of representing bandlimited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density λ ∞, convolving these one-bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. However, unlike fine-resolution quantization, the accuracy of one-bit quantization is not well-understood. A natural error lower bound that decreases like 2−λ can easily be given using information theoretic arguments. Yet, no one-bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper, we construct an infinite family of one-bit sigma-delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for π-bandlimited signals is at most O(2−.07λ). © 2003 Wiley Periodicals, Inc.
TL;DR: In this article, the authors prove uniqueness and a Holder-type stability of reconstruction of all three time-independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements.
Abstract: We prove uniqueness and a Holder-type stability of reconstruction of all three time-independent elastic parameters in the dynamical isotropic system of elasticity from two special sets of boundary measurements. In proofs we use Carleman-type estimates in Sobolev spaces of negative order. © 2003 Wiley Periodicals, Inc.
TL;DR: The argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining a dissipation relation, an isoperimetric inequality, and an ODE lemma, and the interpolation inequality is new and rather subtle.
Abstract: We study a specific example of energy-driven coarsening in two space dimensions. The energy is ∫|∇∇u|2 + (1 - | ∇u|2)2; the evolution is the fourth-order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1/3, and the energy per unit area decays like t-1/3. We prove a weak, one-sided version of the latter statement: The time-averaged energy per unit area decays no faster than t-1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles-Giga energy. © 2003 Wiley Periodicals, Inc.
TL;DR: The equation of Camassa and Holm [2]2 is an approximate description of long waves in shallow water as mentioned in this paper, and it is used to describe the long wave in shallow waters.
Abstract: The equation of Camassa and Holm [2]2 is an approximate description of long waves in shallow water.
TL;DR: In this paper, the authors established global uniqueness and obtained reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials in dimensions greater than or equal to three, where the potentials considered are conormal of order less than 1-k with respect to submanifolds.
Abstract: In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order less than 1-k with respect to submanifolds (of arbitrary codimension k). This gives positive results for (conormal) conductivities which are Holder of any order > 1. A related problem for highly singular potentials is shown to exhibit nonuniqueness.
[...]
TL;DR: In this paper, the question of well-posedness of the Cauchy problem for Schrodinger maps from ℝ ×ℝ2 to the sphere 2 or to ℍ2, the hyperbolic space was studied.
Abstract: We study the question of well-posedness of the Cauchy problem for Schrodinger maps from ℝ × ℝ2 to the sphere 2 or to ℍ2, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrodinger system of equations and then study this modified Schrodinger map system (MSM). We then prove local well-posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well-posedness of the Schrodinger map itself from it. In proving well-posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L2-functions. © 2002 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors study the problem of the motion of the free surface of a liquid and prove existence and stability for the linearized equations for the nonlinearized equations.
Abstract: We study the problem of the motion of the free surface of a liquid. We prove existence and stability for the linearized equations.
TL;DR: In this article, the authors studied the topological entropy of a nonlinear reaction-diffusion system in an unbounded domain and showed that it possesses a global attractor in the corresponding phase space.
Abstract: The nonlinear reaction-diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ϵ-entropy. Upper and lower bounds of this entropy are obtained.
Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ℝn. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space . It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy.
In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz-continuous) embedding of this system into the spatial shifts on the attractor.
Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.
TL;DR: In this article, the existence of boundary traces for positive solutions of the equation −Δu + g(x, u) = 0 in a smooth domain Ω ⊂ ℝN, for a general class of positive nonlinearities was established.
Abstract: In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation −Δu + g(x, u) = 0 in a smooth domain Ω ⊂ ℝN, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller-Osserman condition as well as degenerate nonlinearities gα,q of the form gα,q (x, u) = d(x, ∂Ω)α |u|q−1u, with α > −2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {gα,q}, determine the critical value of the exponent q (for fixed α > −2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for −Δu + gα,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.
TL;DR: In this article, the authors continue the recent study of an Allen-Cahn model PDE by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions, enabling them to obtain multibump solutions in a much more general setting.
Abstract: This paper continues the recent study of an Allen-Cahn model PDE [1] by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions, enabling us to obtain multibump solutions in a much more general setting. As in [1], novel minimization arguments play a key role in finding solutions.
TL;DR: In this paper, it was shown that the Galerkin truncation of the Burgers-Hopf equation is a Hamiltonian system with Hamiltonian given by the integral of the third power.
Abstract: In dynamical systems with intrinsic chaos, many degrees of freedom, and many conserved quantities, a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes. The Galerkin truncation of the Burgers-Hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments. Here it is established that the truncated Burgers-Hopf model is a Hamiltonian system with Hamiltonian given by the integral of the third power. This additional conserved quantity, beyond the energy, has been ignored in previous statistical mechanics studies of this equation. Thus, the question arises of the statistical significance of the Hamiltonian beyond that of the energy. First, an appropriate statistical theory is developed that includes both the energy and Hamiltonian. Then a convergent Monte Carlo algorithm is developed for computing equilibrium statistical distributions. The probability distribution of the Hamiltonian on a microcanonical energy surface is studied through the Monte-Carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the Hamiltonian. Empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the Hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy. The predictions of the theory for relevant and irrelevant values for the Hamiltonian are confirmed through systematic numerical simulations. For statistically relevant values of the Hamiltonian, these simulations show a surprising spectral tilt rather than equipartition of energy. This spectral tilt is predicted and confirmed independently by Monte Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt. On the other hand, the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the Hamiltonian with excellent accuracy. The results established here for the Burgers-Hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications. Several interesting mathematical problems suggested by this study are mentioned in the final section. © 2002 Wiley Periodicals, Inc.
TL;DR: In this article, the authors studied the curvelike structure of special measures on ℝn in a multiscale fashion and constructed a sufficiently short curve with a sufficiently large measure.
Abstract: We study the curvelike structure of special measures on ℝn in a multiscale fashion. More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure. Our main tool is an L2 variant of Jones' β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations. The Jones function is formed by adding the squares of the L2 Jones numbers at different scales and the same location. Using a special L2 Jones function, we construct a sufficiently short curve with a sufficiently large measure. The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function. © 2003 Wiley Periodicals, Inc.