The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003     

Laurent expansions for certain functions defined by Dirichlet series

Summary The Poisson summation formula is employed to find the Laurent expansions of the Dirichlet seriesF(s, c) = _{n = 0} exp[–(n + c)^1/2 s] andG(s, c) = _{n = 0} (–1) ⁿ exp[–(n + c)^1/2 s] (0c<1) abouts = 0. The Laurent expansions ofF(s, c) andG(s, c) are convergent respectively for 0 < |s| < and |s| < , and define the analytic continuation of the Dirichlet series to the half-plane Res < 0.     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

Sonar echo analysis

Pulse mode sonar operation is analyzed under the assumption that the scattering object Γ lies in the far fields of both the transmitter and the receiver. It is shown that, in this approximation, the sonar signal is a plane wave s(x · θ₀ – t) near Γ, where θ₀ is a unit vector directed from the transmitter toward Γ, and similarly the echo is a plane wave e(x · θ – t) near the receiver, where θ is a unit vector directed from Γ toward the receiver. Moreover, if Γ is stationary with respect to the sonar system then it is shown that where ?(ω) is the Fourier transform of S(τ) and T+(ωθ,ωθ₀) is the scattering amplitude in the direction θ due to the scattering by Γ of a time-harmonic plane wave with frequence ω and propagation direction θ₀. A generalization of this relation is derived for moving scatterers.     

Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u_t = ? ? (?(u) ? u) ? χ ? ? (u ? v) + g(u), ? Δv = ? v + u in Ω × (0,T), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain with smooth boundary, n ≥ 1, χ > 0, ? ≥ c₁s^p for s ≥ s₀ > 1, and g(s) ≤ as ? μs² for s > 0 with a,g(0) ≥ 0, μ > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever , or, equivalently, , which enlarge the parameter range , or , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999     

On the effect of external forces on incompressible fluid motions at large distances

We study incompressible Navier–Stokes flows in \mathbb R^d{\mathbb R^d} with small and well localized data and external force f. We establish pointwise estimates for large |x| of the form c_t|x|^-d ￡ |u(x,t)| ￡ c￠_t|x|^-d{c_t|x|^{-d}\le |u(x,t)|\le c^\prime_t|x|^{-d}}, where c _t > 0 whenever ò₀^tòf(x,s) dx ds 1 0{\int_0^t\int f(x,s)\,dx\,ds\not= {\bf 0}} . This sharply contrasts with the case of the Navier–Stokes equations without force, studied in Brandolese and Vigneron (J Math Pures Appl 88:64–86, 2007) where the spatial asymptotic behavior was |u(x,t)| @ C_t|x|^-d-1{|u(x,t)|\simeq C_t|x|^{-d-1}} . In particular, this shows that external forces with non-zero mean, no matter how small and well localized (say, compactly supported in space-time), increase the velocity of fluid particles at all times t and at at all points x in the far-field. As an application of our analysis on the pointwise behavior, we deduce sharp upper and lower bounds of weighted L ^p -norms for strong solutions, extending the results obtained in Bae et al. (to appear) for weak solutions, by considering here a larger (and in fact, optimal) class of weight functions.     

Nonradial Large Solutions of Sublinear Elliptic Equations

We show that the equation Δu = p(x)f(u) has a positive solution on R ^N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup _{s ≥ 1} f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial.     

Isolated singularities of some semilinear elliptic equations

Let Ω be an open subset of $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">N</mi>}}$, N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|^{1 ? N}) we prove that u is isotropic near 0 and show that either it is a C¹ function in Ω (removable singularity) or |x|^{N ? 2}u(x) → c, c ≠ 0 (weak singularity) or |x|^{N ? 2} |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166).     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

An Asymptotic Analysis of Localized Solutions for Some Reaction-Diffusion Models in Multidimensional Domains

In the limit ? → 0, a spike-layer solution is constructed for the reaction-diffusion equation where b > 0 and D is a bounded convex domain. Here Q(u) is such that there exists a unique radially symmetric function u_c(?^?1 r) satisfying ?²Δu_c + Q(u_c) = 0 in all of ?^N, with u_c(ρ) decaying exponentially at infinity. The spike-layer solution has the form u ~ u_c [?|x ? x₀|], where the spike-layer location x₀ ? D is to be determined subject to the condition that dist(x₀, ?D) as ? → D. The determination of x₀ is shown to be exponentially ill conditioned and asymptotic estimates for the exponentially small eigenvalues and the corresponding eigenfunctions associated with the linearized problem are obtained. These spectral results are used together with a limiting solvability condition to derive an equation for x₀. For a strictly convex domain, it is shown that there is an x₀ that is located at an O(?) distance away from the point in D that is furthest from ?D. Finally, hot-spot solutions to Bratu's equation are constructed asymptotically in a singularly perturbed limit.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Moser's second one-dimensional inequality for a class of integral operators

Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? L^p(-￥ ,￥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò₀^xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò_-￥^￥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant C_pC_p, such that ò_-￥^￥ exp[a |F(x)|^q-|x|]  dx ￡ C_p\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||_p ￡ 1||f||_p\le 1 and every a ￡ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.     

Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:

• u ≥ φ in ?ⁿ,
• (??)^su ≥ 0 in ?ⁿ,
• (??)^su(x) = 0 for those x such that u(x) > φ(x),
• lim_{|x| → + ∞} u(x) = 0.

We show that when φ is C^{1, s} or smoother, the solution u is in the space C^{1, α} for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C^{1, s}. When φ is only C^{1, β} for a β < s, we prove that our solution u is C^{1, α} for every α < β. © 2006 Wiley Periodicals, Inc.