Positive solutions for Robin problem involving the -Laplacian

Consider Robin problem involving the p(x)-Laplacian on a smooth bounded domain Ω as follows

Applying the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that there exists λ_*>0 such that the problem has at least two positive solutions if λ(0,λ_*), has at least one positive solution if λ=λ_*<+∞ and has no positive solution if λ>λ_*. To prove the results, we prove a norm on W^1,p(x)(Ω) without the part of ||_L^p(x)(Ω) which is equivalent to usual one and establish a special strong comparison principle for Robin problem.     

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation

with p>p_S where p_S is the Sobolev exponent. If u(x,t)=(T−t)^−1/(p−1)φ((T−t)^−1/2x) for xR^N and t[0,T), where φ is a regular positive solution of

(P)

then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)^−1/(p−1) for all p>1. Let p_L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p_S<p<p_L. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>p_L.     

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

Semilinear equations with exponential nonlinearity and measure data

We study the existence and non-existence of solutions of the problem

(0.1)

where Ω is a bounded domain in , N3, and μ is a Radon measure. We prove that if , then (0.1) has a unique solution. We also show that the constant 4π in this condition cannot be improved.     

Traveling fronts in space–time periodic media

This paper is concerned with the existence of pulsating traveling fronts for the equation:

(1)

∂_tu−(A(t,x)u)+q(t,x)u=f(t,x,u),

where the diffusion matrix A, the advection term q and the reaction term f are periodic in t and x. We prove that there exist some speeds c^* and c^** such that there exists a pulsating traveling front of speed c for all cc^** and that there exists no such front of speed c<c^*. We also give some spreading properties for front-like initial data. In the case of a KPP-type reaction term, we prove that c^*=c^** and we characterize this speed with the help of a family of eigenvalues associated with the equation. If f is concave with respect to u, we prove some Lipschitz continuity for the profile of the pulsating traveling front.     

Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

In this article, we consider the following eigenvalue problems

('∗

λ' render=n">

where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ(0,λ^*), a unique positive solution if λ=λ^*, and no positive solution if λ>λ^*. We also obtain some further properties of the positive solutions of (*)_λ.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

Asymptotic behavior of Solutions for Hénon systems with nearly critical exponent

We consider in this paper the problem

(0.1)

where Ω is the unit ball in centered at the origin, 0α<pN, β>0, N8, p>1, q_ε>1. Suppose q_ε→q>1 as ε→0⁺ and q_ε,q satisfy respectively

we investigate the asymptotic behavior of the ground state solutions (u_ε,v_ε) of (0.1) as ε→0⁺. We show that the ground state solutions concentrate at a point, which is located at the boundary. In addition, the ground state solution is non-radial provided that ε>0 is small.     

Existence of multiple positive solutions for nonlinear m-point boundary-value problems

In this paper, we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m-point boundary-value problem for the one-dimensional p-Laplacian

(φ_p(u^′))^′+f(t,u)=0, t(0,1),

     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

In this paper we study the nonlocal p-Laplacian type diffusion equation,

If p>1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation u_t=div(|u|^p−2u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L^∞(0,T;L^p(Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p=1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

L r Estimates for Degenerate Elliptic Problems

We prove L ^r estimates for the Dirichlet problem –div(a(x,u,Du))=f with f in L ^q for 1q+, where the operator satisfies (|s|)|| ^p a(x,s,), with p>1. These estimates are obtained without symmetrization and are sharp in some cases.     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems

In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type

where Ω is a bounded domain in with smooth boundary, 1<p<n,Δ_pu=div(|u|^p-2u) is the p-Laplacian operator, , , (x)0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.     

Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

On the relaxation of nonconvex superficial integral functionals

We present a new approach to the variational relaxation of functionals of the type:

where is a continuous function with growth conditions of order p≥1 but not necessarily convex. We essentially study the case when μ is the k-dimensional Hausdorff measure restricted to a suitable piece of a k-dimensional smooth submanifold of .     

Curves of positive solutions of boundary value problems on time-scales

Let TR be a time-scale, with a=infT, b=supT. We consider the nonlinear boundary value problem

(2)

(4)

u(a)=u(b)=0,