Principal Eigenvalues and Anti – Maximum Principle for Some Quasilinear Elliptic Equations on ℝN

We improve some previous existence and nonexistence results for positive principal eigenvalues of the problem —Δ_pu = λg(x)ψ^p(u), x ∈ ℝ^N, lim_{‖x‖⇒+∞}u(x) = 0. Also we discuss existence, nonexistence and antimaximum principle questions concerning the perturbed problem —Δ_pu = λg(x)ψ^p(u) + f(x), x∈ ℝ^N.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

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.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u² in a domain D of class C² of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, v) where K is a closed subset of ∂D and v is a Radon measure on ∂D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.     

Single‐cell discretization of O(kh2 + h4) for ∂u/∂n for three‐space dimensional mildly quasi‐linear parabolic equation

In this article, using a single computational cell, we report some stable two‐level explicit finite difference approximations of O(kh² + h⁴) for ?u/?n for three‐space dimensional quasi‐linear parabolic equation, where h > 0 and k > 0 are mesh sizes in space and time directions, respectively. When grid lines are parallel to x‐, y‐, and z‐coordinate axes, then ?u/?n at an internal grid point becomes ?u/?x, ?u/?y, and ?u/?z, respectively. The proposed methods are also applicable to the polar coordinates problems. The proposed methods have the simplicity in nature and use the same marching type of technique of solution. Stability analysis of a linear difference equation and computational efficiency of the methods are discussed. The results of numerical experiments are compared with exact solutions. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 327–342, 2003.     

On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

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.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity

In this paper, we are concerned with the existence of solutions to the N-dimensional nonlinear Schrödinger equation −ε²Δu+V(x)u=K(x)up with u(x)>0, u∈H¹(RN), N?3 and . When the potential V(x) decays at infinity faster than ⁻²(1+|x|) and K(x)?0 is permitted to be unbounded, we will show that the positive H¹(RN)-solutions exist if it is assumed that G(x) has local minimum points for small ε>0, here with denotes the ground energy function which is introduced in [X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633-655]. In addition, when the potential V(x) decays to zero at most like (1+|x|)^−α with 0<α?2, we also discuss the existence of positive H¹(RN)-solutions for unbounded K(x). Compared with some previous papers [A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006) 317-348; A. Ambrosetti, D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907; A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321-1332] and so on, we remove the restrictions on the potential function V(x) which decays at infinity like (1+|x|)^−α with 0<α?2 as well as the restrictions on the boundedness of K(x)>0. Therefore, we partly answer a question posed in the reference [A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, preprint, 2006].     

Formula for a solution ofu t +H(u,Du) =g

We study the continuous as well as the discontinuous solutions of Hamilton-Jacobi equationu _t +H(u,Du) =g in ℝ ⁿ x ℝ₊ withu(x, 0) =u ₀(x). The HamiltonianH(s,p) is assumed to be convex and positively homogeneous of degree one inp for eachs in ℝ. IfH is non increasing ins, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.     

Asymptotic behaviour of a Brownian motion on exterior domains

We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ? ^c is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p ^? _t (x, y) behaves, for large t, as u(x)u(y)(t(log t)²)⁻¹ for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?) ^r , such that u(x) ∼ log ∣x∣. Received: 29 January 1999 / Revised version: 11 May 1999     

On pointwise convergence of the family of Urysohn‐type integral operators

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L₁?function. The kernel K_λ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and K_λ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L₁(R).     

On singular perturbations of superlinear elliptic systems

We study the existence, multiplicity and shape of positive solutions of the system −ε²Δu+V(x)u=K(x)g(v), −ε²Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous.     

含频散项的K(2,3)方程的精确行波解

本文研究了包含频散项的K(2,3)方程u_t+(u²)_x-(u³)_xxx=0的分支问题.利用动力系统的定性分析,并且借助Maple软件进行数值模拟得到行波解系统相应的相图,然后通过积分计算得到周期尖波解、类扭波和类反扭波的精确解的函数表达式,以及孤立波精确解的隐函数表达式.     

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)u_t=(g(x)uⁿu_x)_x+h(x)u^m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.     

On the existence of shock fronts for a Burgers equation with a singular source term

In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u⁰₋(x) for x < x₀, u(x, 0) = u⁰₊(x) for x > x₀, u⁰₋(x₀) > u⁰₊(x₀). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u⁰₋ and u⁰₊, a global shock front weak solution u(x, t) = u₋(x, t) for x < ϕ(t), u(x, t) = u₊(x, t) for x > ϕ(t), where u₋ and u₊ are the strong solutions corresponding (respectively) to u⁰₋ and u⁰₊ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u₋(ϕ(t), t) + u₊(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x₀. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.