Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f ⁻¹(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

Geodesic flow on three-dimensional ellipsoids with equal semi-axes

Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with SO(2) × SO(2) symmetry, ellipsoids with equal larger or smaller semiaxes with SO(2) symmetry, and ellipsoids with three semi-axes coinciding with SO(3) symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with SO(2) symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with SO(2) × SO(2) symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with SO(3) symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are T ² bundles over S ².      

Marginal relevance of disorder for pinning models

The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched models have different critical points and critical exponents) if the return probability exponent α, a positive number that characterizes the model, is larger than ½. Weak disorder has been predicted to be irrelevant (i.e., coinciding critical points and exponents) if α < ½. Recent mathematical work has put these predictions on firm ground. In renormalization group terms, the case α = ½ is a marginal case, and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is also particularly intriguing because the case α = ½ includes the classical models of two‐dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3 + 1) or (1 + 1), and of pinning of an heteropolymer by a point potential in three‐dimensional space. Here we prove disorder relevance both for the general α = ½ pinning model and for the hierarchical pinning model proposed by Derrida, Hakim, and Vannimenus, in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(?1/β⁴) for β small, if β² is the disorder variance. © 2009 Wiley Periodicals, Inc.     

Existence of global smooth solutions for euler equations with symmetry

Eventhough existence of global smooth solutions for one dimensional quasilinear hyperbolic systems has been well established, much less is known about the corresponding results for higher dimensional cases. In this paper, we study the existence of global smoothe solutions for the initial-boundary value problem ofo Euler equtions satisfying γ law with damping and exisymmetry, or spherical symmetry. When the damping is strong enough, we give some sufficient conditions for existence of global smooth solutions as 1<γ< 5 3 and 5 3 <γ<3 . The proof is based on technical estimation of the C ¹ norm of the solutions.     

Repellers for non-uniformly expanding maps with singular or critical points

GivenanergodicmeasurewithpositiveentropyandonlypositiveLyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps are studied that are C ^1+β smooth outside a set of possibly critical or singular points.     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Statistical manifolds are statistical models

In this note we prove that any smooth (C¹ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers to Lauritzen’s question and Amari’s question on a realization of smooth (C¹ resp.) statistical manifolds as finite dimensional statistical models.     

On Absorbing Cycles in Min–Max Digraphs

We consider smooth finite dimensional optimization problems with a compact, connected feasible set M and objective function f. The basic problem, on which we focus, is: how to get from one local minimum to all the other ones. To this aim we introduce a bipartite digraph as follows. Its nodes are formed by the set of local minima and maxima of f|_M, respectively. Given a smooth Riemannian (i.e. variable) metric, there is an arc from a local minimum x to a local maximum y if the ascent (semi-)flow induced by the projected gradients of f connects points from a neighborhood of x with points from a neighborhood of y. The existence of an arc from y to x is defined with the aid of the descent (semi-)flow. Strong connectedness of ensures that, starting from one local minimum, we may reach any other one using ascent and descent trajectories in an alternating way. In case that no inequality constraints are present or active, it is well known that for a generic Riemannian metric the resulting min-max digraph is indeed strongly connected. However, if inequality constraints are active, then there might appear obstructions. In fact, we show that may contain absorbing two-cycles. If one enters such a cycle, one cannot leave it anymore via ascent and descent trajectories. Moreover, the cycles being constructed are stable with respect to small perturbations (in the C¹-topology) of the Riemannian metric.     

Convergence of Equilibria of Thin Elastic Plates – The Von Kármán Case

We study the behaviour of thin elastic bodies of fixed cross-section and of height h, with h → 0. We show that critical points of the energy functional of nonlinear three-dimensional elasticity converge to critical points of the von Kármán functional, provided that their energy per unit height is bounded by Ch ⁴ (and that the stored energy density function satisfies a technical growth condition). This extends recent convergence results for absolute minimizers.     

Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds

For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N ^{n +1},g), with non-trivial n dimensional integer homology, such that for some Γ∈H _n (N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities. Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998     

Counting the onion

Iteratively computing and discarding a set of convex hulls creates a structure known as an “onion.” In this paper, we show that the expected number of layers of a convex hull onion for n uniformly and independently distributed points in a disk is Θ(n^2/3). Additionally, we show that in general the bound is Θ(n^2/(d+1)) for points distributed in a d‐dimensional ball. Further, we show that this bound holds more generally for any fixed, bounded, full‐dimensional shape with a nonempty interior. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Finite element methods for semilinear parabolic interface problems

In this exposition we study the finite element methods for second-order semilinear parabolic interface problems in two dimensional convex polygonal domains with smooth interface. Both semidiscrete and fully discrete schemes are analyzed. Optimal order error estimates in the L²(0, T; H¹(Ω))-norm are established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

An elementary proof of a theorem of Johnson and Lindenstrauss

A result of Johnson and Lindenstrauss [13] shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/?²)‐dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 ± ?). In this note, we prove this theorem using elementary probabilistic techniques. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 60–65, 2002     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

On the regularity of spherically symmetric wave maps

Wave maps are critical points U: M → N of the Lagrangian ??[U] = ∞_M ‖dU‖², where M is an Einsteinian manifold and N a Riemannian one. For the case M = ?^2,1 and U a spherically symmetric map, it is shown that the solution to the Cauchy problem for U with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded; and (2) there are two constants c and C such that the smallest eigenvalue λ and the largest eigenvalue λ of the second fundamental form k_AB of any geodesic sphere Σ(p, s) of radius s centered at p ? N satisfy sλ ≧ c and s A ≦ C(1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. © 1993 John Wiley & Sons, Inc.     

Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.