Some multipoint boundary value problems of Neumann-Dirichlet type involving a multipoint p-Laplace like operator

Let ? and θ be two increasing homeomorphisms from R onto R with ?(0)=0, θ(0)=0. Let be a function satisfying Carathéodory's conditions, and for each i, i=1,2,…,m−2, let , be a continuous function, with , ξi∈(0,1), 0<ξ₁<ξ₂<?<ξm−2<1.In this paper we first prove a suitable continuation lemma of Leray-Schauder type which we use to obtain several existence results for the m-point boundary value problem:

     

Growth rate of the functions in Bergman type spaces

For 0<p,α<∞, let ‖f‖_p,α be the L^p-norm with respect the weighted measure . We define the weighted Bergman space A_α^p(D) consisting of holomorphic functions f with ‖f‖_p,α<∞. For any σ>0, let A^−σ(D) be the space consisting of holomorphic functions f in D with . If D has C² boundary, then we have the embedding A_α^p(D)⊂A^−(n+α)/p(D). We show that the condition of C²-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C^1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u^″+f(u) defined on a Sobolev space of periodic functions Hp(S¹), p?1. Let be the plane z=0 and, for n>0, let n be the cone x²+y²=tan²z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S¹),C) and (R³,Σ)×H where H is a real separable infinite-dimensional Hilbert space.     

An improved local well-posedness result for the one-dimensional Zakharov system

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schrödinger data and wave data under certain assumptions on the parameters k,l and 1<p?2, where , generalizing the results for p=2 by Ginibre, Tsutsumi and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<−1/2, i.e. data u₀∉L² and (n₀,n₁)∉H−1/2×H−3/2, which was excluded in the case p=2.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Exponential stability of nonautonomous linear differential equations with linear perturbations by Liao methods

In this paper, the author considers, by Liao methods, the stability of Lyapunov exponents of a nonautonomous linear differential equations: with linear small perturbations. It is proved that, if A(t) is a upper-triangular real n by n matrix-valued function on R₊, continuous and uniformly bounded, and if there is a relatively dense sequence in R₊, say 0=T₀<T₁<?<Ti<?, such that

     

Two results on homogeneous Hessian nilpotent polynomials

Let z=(z₁,…,z_n) and , the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix is nilpotent. In recent developments in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (8) (2005) 2201-2205. [MR2138860]; G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (6) (2006) 503-510. [MR2170971]. See also math-ph/0308035; W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249-274. [MR2247890]. See also math.CV/0409534], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture (VC) of HN polynomials: for any homogeneous HN polynomialP(z) (of degreed=4), we haveΔ^mP^m+1(z)=0for anym?0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties Z_P and Z_σ2 of CPⁿ⁻¹ determined by the principal ideals generated by P(z) and , respectively, intersect only at regular points of Z_P. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z−∇P with P(z) HN if F has no non-zero fixed point w∈Cⁿ with . Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), Δ^m(f(z)P(z)^m)=0 when m?0.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Differential operator specializations of noncommutative symmetric functions

Let K be any unital commutative Q-algebra and z=(z₁,…,zn) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K《z》 and K?t?《z》 the formal power series algebras of z over K and K?t?, respectively. For any α?1, let D[α]《z》 be the unital algebra generated by the differential operators of K《z》 which increase the degree in z by at least α−1 and the group of automorphisms Ft(z)=z−Ht(z) of K?t?《z》 with o(Ht(z))?α and Ht=0(z)=0. First, for any fixed α?1 and , we introduce five sequences of differential operators of K《z》 and show that their generating functions form an NCS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363-382] over the differential algebra D[α]《z》. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms , which are also grading-preserving when Ft satisfies certain conditions. Note that the homomorphisms SFt above can also be viewed as specializations of NCSFs by the differential operators of K《z》. Secondly, we show that, in both commutative and noncommutative cases, this family SFt (with all n?1 and ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289-301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967-982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved

     

Solvability of singular second order m-point boundary value problems

Let be a function satisfying Carathéodory's conditions and (1−t)e(t)∈L¹(0,1). Let ξi∈(0,1), ai∈R, i=1,…,m−2, 0<ξ₁<ξ₂<?<ξm−2<1 be given. This paper is concerned with the problem of existence of a C¹[0,1) solution for the m-point boundary value problem