Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

The red‐shift effect and radiation decay on black hole spacetimes

We consider solutions to the linear wave equation □_g? = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of ? may contain the bifurcate event horizon.) It is shown that the energy flux F(??) of the solution (as measured by a strictly timelike T? that asymptotically matches the static Killing field) through arbitrary achronal subsets ?? of the black hole exterior region satisfies the bound F(??) ≤ C E(v + u), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of ??, v₊ denotes max{1, v}, and u₊ denotes max{1, u}, where C is a constant depending only on the parameter M, and E depends on a suitable norm of the solution on the hypersurface t ? u + v = 1. (The bound applies in particular to subsets ?? of the event horizon or null infinity.) It is also shown that ? satisfies the pointwise decay estimate |?| ≤ C Ev in the entire exterior region, and the estimates |r?| ≤ C_R?E(1 + |u|)^?1/2 and |r^1/2?| ≤ C_R?Eu in the region {r ≥ R?} ∩ J⁺(Σ) for any R? > 2M. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red‐shift effect. The results in particular give an independent proof of the classical result |?| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.     

On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

Star factorizations of graph products

A k‐star is the graph K_1,k. We prove a general theorem about k‐star factorizations of Cayley graphs. This is used to give necessary and sufficient conditions for the existence of k‐star factorizations of any power (K_q)^s of a complete graph with prime power order q, products C × C ×··· × C of k cycles of arbitrary lengths, and any power (C_r)^s of a cycle of arbitrary length. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 59–66, 2001     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

Law of large numbers for increasing subsequences of random permutations

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1,…,n}. We show that Var(Z) = o((EZ)²) as n → ∞ if and only if . In particular then, the weak law of large numbers holds for Z if ; that is, We also show the following approximation result for the uniform measure U_n on S_n. Define the probability measure μ on S_n by where U denotes the uniform measure on the subset of permutations that contain the increasing subsequence {x₁,x₂,…,x}. Then the weak law of large numbers holds for Z if and only if where ∣∣˙∣∣ denotes the total variation norm. In particular then, (*) holds if . In order to evaluate the asymptotic behavior of the second moment, we need to analyze occupation times of certain conditioned two‐dimensional random walks. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Iteratively reweighted least squares minimization for sparse recovery

Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ?^N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ^?1(y) is typically an (N ? m)—dimensional hyperplane; in addition, x is then equal to the element in Φ^?1(y) of minimal ??₁‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ^?1(y) with smallest ??₂(w)‐norm. If x⁽ⁿ⁾ is the solution at iteration step n, then the new weight w⁽ⁿ⁾ is defined by w := [|x|² + ε]^?1/2, i = 1, …, N, for a decreasing sequence of adaptively defined ε_n; this updated weight is then used to obtain x^{(n + 1)} and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x⁽ⁿ⁾ converges for all y, regardless of whether Φ^?1(y) contains a sparse vector. If there is a sparse vector in Φ^?1(y), then the limit is this sparse vector, and when x⁽ⁿ⁾ is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [|x|² + ε]^?1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc.     

Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

A quantitative compactness estimate for scalar conservation laws

In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math. 7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ?‐entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L¹. © 2005 Wiley Periodicals, Inc.     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

Reversible and DRAD difference sets in

The existence of Hadamard difference sets has been a central question in design theory. Reversible difference sets have been studied extensively. Dillon gave a method for finding reversible difference sets in groups of the form (C)². DRAD difference sets are a newer concept. Davis and Polhill showed the existence of DRAD difference sets in the same groups as Dillon. This article determines the existence of reversible and DRAD difference sets in groups of the form (C)³. These are the only abelian 2‐groups outside of direct products of C₄ and (C)² known to contain reversible and DRAD difference sets. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:58–67, 2012     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

The Calderón approach to an elliptic boundary problem

We consider a boundary problem for an elliptic system in a bounded region Ω ? ?ⁿ and where the spectral parameter is multiplied by a discontinuous weight function ω (x) = diag(ω₁(x), …, ω_N (x)). The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Recently, this problem was studied under the assumption that the ω_j (x)^–1 are essentially bounded in Ω. In this paper we suppose that ω (x) vanishes identically in a proper subregion Ω of Ω and that the ω_j (x)^–1 are essentially bounded in . Then by using methods which are a variant of those used in constructing the Calderón projectors for the boundary Γ of Ω, we shall derive results here which will enable us in a subsequent work to apply the ideas of Calderón to develop the spectral theory associated with the problem under consideration here (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Highly oscillatory multidimensional shocks

We construct geometric optics expansions of high order for oscillatory multidimensional shocks and then show that the expansions are close to exact shock solutions for small wavelengths. Expansions are constructed both for S_ϵ, the oscillatory function defining the shock surface S_ϵ, and for u, the solutions on each side of S_ϵ. The profile equations yield detailed information on the evolution of (u, ψ_ϵ), showing, for example, how new interior oscillations are produced by a variety of shock—interior and interior—interior interactions. A generic small divisor property, L²‐estimates for linearized shock problems with merely Lipschitz coefficients, and a continuation principle based on an unusual Gagliardo‐Nirenberg inequality all play a role in the proofs. © 1999 John Wiley & Sons, Inc.     

A new construction of perfect nonlinear functions using Galois rings

For a prime p, we give a construction of perfect nonlinear functions from ? to ? when either of the following conditions holds: (1) n≥p; (2) n<p, and n is a composite number or is the sum of positive composite numbers. It follows that when n≥12, there is a perfect nonlinear function from ? to ? for any prime p. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 229‐239, 2009     

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|² : ∇u ∈ L²(R₊ⁿ), ∫ |u|^q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫_dM|u|^q ds_g)^2/q < or = to S ∫_M |∇_gu|² dv_g + A ∫_dMu² ds_g, for all u ∈ H¹ (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc.     

Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈ H^1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H¹(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Edge-colored saturated graphs

A graph G is (k₁, k₂, …, k_t)-saturated if there exists a coloring C of the edges of G in t colors 1, 2, …, t in such a way that there is no monochromatic complete k_i-subgraph K of color i, 1 ? i ? t, but the addition of any new edge of color i, joining two nonadjacent vertices in G, with C, creates a monochromatic K of color i, 1 ? i ? t. We determine the maximum and minimum number of edges in such graphs and characterize the unique extremal graphs.     

Convergence of boundary optimal controls in mixed elliptic problems

We consider a steady-state heat conduction problem P_α withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ₁ of the boundary of Ω. We consider, for each α > 0, a cost function J_α and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ₂ of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ₂ in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to q_op, u and p in L²(Γ₂), H¹(Ω), and H¹(Ω) respectively when α → ∞. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition on the portion Γ₁. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux q defined on Γ₂ and the parameter α (which goes to infinity) is defined on Γ₁. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow‐up estimates for a quasi‐linear reaction–diffusion system

In this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣?_u∣^p‐2?_u) = uv, ‐div(∣?_u∣^q‐2?_u) = uv in ?^N(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems u_t = div (∣?_u∣^p‐2?_u)+uv,vt = div(∣?_v∣^q‐2?_v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.