A new approach to variational problems with multiple scales

We introduce a new concept, the Young measure on micropatterns, to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter ε. This allows one to extract, in the limit ε → 0, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say ε^α) and to eliminate all finer scales. To achieve this we consider rescaled functions Rx (t) := x (s + ε^αt) viewed as maps of the macroscopic variable s ∈ Ω with values in a suitable function space. The limiting problem can then be formulated as a variational problem on the Young measures generated by R^εx. As an illustration, we study a one‐dimensional model that describes the competition between formation of microstructure and highest gradient regularization. We show that the unique minimizer of the limit problem is a Young measure supported on sawtooth functions with a given period. © 2001 John Wiley & Sons, Inc.     

Eigenvalue Degeneracy and Existence and Non-existence of a Phase Transition in One Dimension

We consider a one dimensional Ising chain with interaction potential J(k) such that J(k) = 0 when k > n. By a perturbation argument we show that long range order exists at sufficiently low temperatures if and only if This is consistent with Dyson's recent theorems and in addition predicts that when J(k) = k^?2 there is no long range order.     

Superdiffusions and removable singularities for quasilinear partial differential equations

To every second-order elliptic differential operator L and to every number α ϵ (1, 2] there is a corresponding measure-valued Markov process X called the (L, α)-superdiffusion. Suppose that Γ is a closed set in R^d. It is known that the following three statements are equivalent: (α) the range of X does not hit Γ; (β) if u ≥ 0 and Lu = u^α in R^d\Γ, then u = 0 (in other words, Γ is a removable singularity for all solutions of equation Lu = u^α); (γ) Cap_2,α′(Γ) = 0 where 1/α + 1/α′ = 1 and Cap_γ,q is the so-called Bessel capacity. The equivalence of (β) and (γ) was established by Baras and Pierre in 1984 and the equivalence of (α) and (β) was proved by Dynkin in 1991. In this paper, we consider sets Γ on the boundary ∂D of a bounded domain D and we establish (assuming that ∂D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Γ (b) if u ≥ 0 and Lu = u^α in D, and if u → 0 as x → α ϵ ∂D\Γ, then u = 0; (c) Cap^∂_2/α,α′(Γ) = 0 where Cap^∂_γ-qis the Bessel capacity on ∂D. This implies positive answers to two conjectures posed by Dynkin a few years ago. (The conjectures have already been confirmed for α = 2 and L = Δ in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivalence of (a)-(c) but also give a new, simplified proof of the equivalence of (α)-(γ). The paper consists of an Introduction (Section 1) and two parts, probabilistic (Sections 2 and 3) and analytic (Sections 4 and 5), that can be read independently. An important probabilistic lemma, stated in the Introduction, is proved in the Appendix. © 1996 John Wiley & Sons, Inc.     

The Theory of Constructive Signal Analysis

Let h(x) = e^?αxk(x), where and λ0=0. The closure theorem, V_h = L¹(?), is proved for various α and k (V_h is the L¹-closed variety generated by h). The Tauberian condition, |?| > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γ_j} generated by {λ_j}. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”     

A nonoscillation theorem for superlinear Emden-Fowler equations with near-critical coefficients

We are interested in the oscillatory behavior of solutions of the Emden-Fowler equation y^″+a(x)|y|^γ−1y=0, γ>1, where a(x) is a positive continuous function on (0,∞). In the special case when the coefficient a(x) is a power of x, i.e. a(x)=xα for some constant α, the value α^∗=−(γ+3)/2 plays a critical role: The equation has both oscillatory and nonoscillatory solutions if α>α^∗, while all solutions are nonoscillatory if α<α^∗. When a(x) is close to the critical exponent, one of the known results is that if a(x)=x−(γ+3)/2log−σ(x), where σ>0, then all solutions are nonoscillatory. In this paper, this result is further extended to include a class of coefficients in which the above condition with log(x) can be replaced by loglog(x), or logloglog(x) and so on.     

Blow‐up analysis,existence and qualitative properties of solutions for the two‐dimensional Emden–Fowler equation with singular potential

Motivated by the study of a two‐dimensional point vortex model, we analyse the following Emden–Fowler type problem with singular potential: where V(x) = K(x)/|x|^2α with α∈(0, 1), 0<a?K(x)?b< + ∞, ?x∈Ω and ∥?K∥_∞?C. We first extend various results, already known in case α?0, to cover the case α∈(0, 1). In particular, we study the concentration‐compactness problem and the mass quantization properties, obtaining some existence results. Then, by a special choice of K, we include the effect of the angular momentum in the system and obtain the existence of axially symmetric one peak non‐radial blow‐up solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Global existence,asymptotic behaviour,and global non‐existence of solutions for damped non‐linear wave equations of Kirchhoff type in the whole space

We study the global existence, asymptotic behaviour, and global non‐existence (blow‐up) of solutions for the damped non‐linear wave equation of Kirchhoff type in the whole space: u_tt+u_t=(a+b∥∇u∥^2γ)Δu+∣u∣^αu in ℝ^N×ℝ₊ for a, b⩾0, a+b>0, γ⩾1, and α>0, with initial data u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x). Copyright © 2000 John Wiley & Sons, Ltd.     

Hope-lax type formular for u∞ t +H u,D u =0: II

The first order equation u _t +H u,D u =0 with u T,x =g x is considered with terminal dat g which is assumed to be only quasiconvex, is a significant generalization of convex functions. The hamiltonian H γ,p is assumed to be homogeneous degree one in p and nondecreasing in γ. It is prove that the explicit solution of such a problem is u t,x = g ^# γ,p T-t H γ,p ^# where # refers to the quasiconvex conjugate of the functions in the x variable.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity

We study the branch of semistable and unstable solutions (i.e., those whose Morse index is at most 1) of the Dirichlet boundary value problem ? Δu = λf(x)/(1 ? u)² on a bounded domain Ω ? ?^N, which models—among other things—a simple electrostatic microelectromechanical system (MEMS) device. We extend the results of 11 relating to the minimal branch, by obtaining compactness along unstable branches for 1 ≤ N ≤ 7 on any domain Ω and for a large class of “permittivity profiles” f. We also show the remarkable fact that powerlike profiles f(x) ? |x|^α can push back the critical dimension N = 7 of this problem by establishing compactness for the semistable branch on the unit ball, also for N ≥ 8 and as long as As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for λ in a natural range. In all these results, the conditions on the space dimension and on the power of the profile are essentially sharp. © 2007 Wiley Periodicals, Inc.     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

The zero dispersion limit of the korteweg-de vries equation for initial potentials with non-trivial reflection coefficient

The inverse scattering method is used to determine the distribution limit as ? → 0 of the solution u(x, t, ?) of the initial value problem. U_t ? 6uu_x + ?²u_xxx = 0, u(x, 0) = v(x), where v(x) is a positive bump which decays sufficiently fast as x x→±α. The case v(x) ? 0 has been solved by Peter D. Lax and C. David Levermore [8], [9], [10]. The computation of the distribution limit of u(x, t, ?) as ? → 0 is reduced to a quadratic maximization problem, which is then solved.     

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.     

On Monge‐Ampère equations with homogenous right‐hand sides

We study the regularity and behavior at the origin of solutions to the two‐dimensional degenerate Monge‐Ampère equation det D²_u = |x|^α with α > ?2. We show that when α > 0, solutions admit only two possible behaviors near the origin, radial and nonradial, which in turn implies C^{2, δ}‐regularity. We also show that the radial behavior is unstable. For α < 0 we prove that solutions admit only the radial behavior near the origin. © 2008 Wiley Periodicals, Inc.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices

We consider a model of long‐range first‐passage percolation on the d‐dimensional square lattice ?d in which any two distinct vertices x,y ? ?^d are connected by an edge having exponentially distributed passage time with mean ‖ x – y ‖^α+o(1), where α > 0 is a fixed parameter and ‖·‖ is the l¹–norm on ?^d. We analyze the asymptotic growth rate of the set ß_t, which consists of all x ? ?^d such that the first‐passage time between the origin 0 and x is at most t as t → ∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α < ^d, (ii) stretched exponential growth for α ? d,2d), (iii) superlinear growth for α ? (2d,2d + 1), and finally (iv) linear growth for α > 2d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α=∞. © 2015 Wiley Periodicals, Inc.     

Picture reconstruction from projections in restricted range

In order to reduce scanning time modern x-ray scanners provide projections only in a restricted range [0, ?] with ? < π. We consider the reconstruction of pictures from p + 1 complete projections in [0, ?]. An extrapolation procedure is given to achieve approximations g_p of the data in the whole range. We show that the L₂-error of the corresponding picture is of order p^?α if the original belongs to the Sobolev space H_o^α. The validity of our error estimate is investigated by numerical experiments.     

Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0

Abstract

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x ², x ², x ²) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x ², x ², x ²) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x ², x ², x ²) = 0.     

Schrödinger operators on regular metric trees with long range potentials: Weak coupling behavior

Consider a regular d-dimensional metric tree Γ with root o. Define the Schrödinger operator −Δ−V, where V is a non-negative, symmetric potential, on Γ, with Neumann boundary conditions at o. Provided that V decays like |x|^−γ at infinity, where 1<γ?d?2, γ≠2, we will determine the weak coupling behavior of the bottom of the spectrum of −Δ−V. In other words, we will describe the asymptotic behavior of infσ(−Δ−αV) as α→0+.