Some properties of semiconcave functions with general modulus

The class of semiconcave functions represents a useful generalization of the one of concave functions. Such an extension can be achieved requiring that a function satisfies a suitable one-sided estimate. In this paper, the structure of the set of points at which a semiconcave function fails to be differentiable—the singular set—is studied. First, we prove some results on the existence of arcs contained on the singular set. Then, we show how these abstract results apply to semiconcave solutions of Hamilton-Jacobi equations.     

A Note on Semiconcave Function

This article gives a simple proof of an equivalent proposition on semiconcave function (see [L.C. Evans (1998). Partial Differential Equations. American Mathematical Society; p. 130]). The proof of sufficiency of the proposition can be easily obtained. We prove its necessity by three steps: First, we prove that the equivalent proposition holds for discrete points <artwork name="GAPA31045ei1">; Secondly, we obtain continuity of semiconcave function; Finally, by using the fact that the sequences λ_m ^k are dense in the interval (0, 1), we prove that the equivalent proposition holds for each λ ∈ (0, 1).     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

Dynamics of Quasi-parabolic One-Resonant Biholomorphisms

In this paper we study the dynamics of germs of quasi-parabolic one-resonant biholomorphisms of ? ⁿ⁺¹ fixing the origin, namely, those germs whose differential at the origin has one eigenvalue 1 and the others having a one-dimensional family of resonant relations. We define some invariants and give conditions which ensure the existence of attracting domains for such maps.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Homoclinic orbits for a class of singular second order Hamiltonian systems in ?3

We consider a conservative second order Hamiltonian system $\ddot q + \nabla V(q) = 0$ in ?³ with a potential V having a global maximum at the origin and a line l ?? {0} = ? as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.     

Exponential asymptotics of bi-oscillatory integrals. Application to the problem of persistence of homoclinic connections for the (iω0)2iω1 resonance

We explain in this Note how to obtain an exponentially small equivalent of a bioscillatory integral when it involves solutions of a nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections for vector fields admitting a (iω₀)²iω₁ resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small bi-oscillatory integral.     

Exponential asymptotics of oscillatory integrals. Application to the problem of persistence of homoclinic connections for the 02iω resonance

We explain in this Note how to obtain an exponentially small equivalent of an oscillatory integral when it involves solutions of nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections to 0 for vector fields admitting a 0²iω resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small oscillatory integral.     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces

In this paper, the multiplicity of Lagrangian orbits on C² smooth compact symmetric star-shaped hypersurfaces with respect to the origin in R2n is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via Z₂-index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425-1436].     

Existence, non-existence and asymptotic behavior of blow-up entire solutions of semilinear elliptic equations

We are mainly concerned with existence, non-existence and the behavior at infinity of non-negative blow-up entire solutions of the equation Δu=ρ(x)f(u) in RN. No monotonicity condition is assumed upon f and, in fact, we obtain solutions with a prescribed behavior both at infinity and at the origin. The method used to get existence is based upon lower and upper solutions techniques while for non-existence we explore radial symmetry, estimates on an associated integral equation and the Keller-Osserman condition.     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

Applications of the group analysis of differential equations to some systems of noncommuting <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth vector fields

Given a canonical basis of C ¹-smooth vector fields \(\{ \tilde X_i \} \) satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to \(\{ \tilde X_i \} \).     

The square of a block is Hamiltonian connected

Let B be a block (finite connected graph without cut-vertices) with at least four vertices and ξ, η be distinct vertices of B. We construct a new block M = M(B, ξ, η) containing five copies of B, and use the existence of a Hamiltonian circuit in M² to establish the existence of a Hamiltonian path starting at ξ and ending at η in B². A variant of this trick shows that B² ? ξ has a Hamiltonian circuit.     

Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L^∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L²-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L^∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.     

Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system

It is proven that if the leftmost eigenvalue is weakly linearly degenerate, then the Cauchy problem for a class of nonhomogeneous quasilinear hyperbolic systems with small and decaying initial data given on a semi-bounded axis admits a unique global C¹ solution on the domain , where x=xn(t) is the fastest forward characteristic emanating from the origin. As an application of our result, we prove the existence of global classical, C¹ solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data given on a semi-bounded axis.     

Holomorphic maps with a given asymptotic expansion at a boundary point

The following result has been known for a long time: let 0 < α < 2π and let S be the sector {z ≠ 0 and arg z ≠ α(+ 2kπ)} of the complex plane; let (u_n) be a given infinite sequence of complex numbers; then there exists a holomorphic function on S which admits the formal power series ∑^+∞_{n = 0}u_nzⁿ as asymptotic expansion at the origin. A first generalization of this result to the infinite dimensional case is given by the author (A result of existence of holomorphic maps which admit a given asymptotic expansion, in “Advances in Holomorphy” (J. A. Barroso, Ed.), in press). We give here an improvement of this last result, based upon a different proof. Then we give two counterexamples showing that our assumptions on the spaces are essential.     

C~∞-solutions for the Second Type of Generalized Feigenbaum's Functional Equations

This work focuses on the second type of generalized Feigenbaum's equation(f(x)) = f(f((x))),f(0) = 1, 0 ≤ f(x) ≤ 1, x ∈ [0, 1],where (x) is C∞-increasing function on [0, 1] and satisfies that (0) = 0, 0 (x) 1(x ∈ [0, 1]).Using constructive method, we discuss the existence of C∞-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation.     

Coordinated search for an object hidden on the line

We consider the coordinated search problem faced by two searchers who start together at zero and can move at speed one to find an object symmetrically distributed on the line. In particular we fully analyze the case of the negative exponential distribution given by the density f(x) = e^−|x|μ/(2μ), μ > 0. The searchers wish to minimize the expected time to find the object and meet back together (with the object) at zero. We give necessary and sufficient conditions for the existence of an optimal search strategy when the target density is continuous and decreasing. We show that for the negative exponential distribution the optimal time is between 4.728μ and 4.729μ. A strategy with expected time in this interval begins with the searchers going in opposite directions and returning to the origin after searching up to successive distances 0.745μ, 2.11μ, 3.9μ, 6μ, 8.4μ,…. These results extend the theory of coordinated search to unbounded regions. It has previously been studied for objects hidden on a circle (by Thomas) and on an interval (by the author).