Proof of some conjectures on the mean-value of Titchmarsh series — III

With some applications in view, the following problem is solved in some special case which is not too special. LetF(s) =Σ _n ^∞ =1a_n λ _n ^−s be a generalized Dirichlet series with 1 =λ ₁ <λ ₂ < …,λ _n≤Dn, andλ _n+1 -λ _n ≥D ^{− 1} λ _n+1 ^{− a} where α>0 andD(≥ 1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for . These results will be applied in other papers to appear later.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Poincaré inequality and Palais–Smale condition for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ₁, where −λ₁ is the first eigenvalue of the Dirichlet p-Laplacian Δ _p on , λ₁ > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ _p associated with −λ₁.     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Supercritical biharmonic equations with power-type nonlinearity

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ² u = |u| ^p-1 u over the whole space , where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p _c, where p _c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p _c. We also study the Dirichlet problem for the equation Δ² u = λ (1 + u) ^p over the unit ball in , where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p _c. Finally, we show that a singular solution exists for some appropriate λ > 0.      

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0,1}ⁿ, where nearest-neighbor edges are occupied with probability p. Let p_c(n) be the value of p for which the expected size of the component containing a fixed vertex attains the value λ2^n/3, where λ is a small positive constant. Let ε=n(p−p_c(n)). In two previous papers, we showed that the largest component inside a scaling window given by |ε|=Θ(2^−n/3) is of size Θ(2^2n/3), below this scaling window it is at most 2(log 2)nε⁻², and above this scaling window it is at most O(ε2ⁿ). In this paper, we prove that for the size of the largest component is at least Θ(ε2ⁿ), which is of the same order as the upper bound. The proof is based on a method that has come to be known as “sprinkling,” and relies heavily on the specific geometry of the n-cube.     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

In this paper we consider the Lane–Emden problem adapted for the p-Laplacian

Convex Minimization over the Fixed Point Set of Demicontractive Mappings

This paper deals with a viscosity iteration method, in a real Hilbert space , for minimizing a convex function over the fixed point set of , a mapping in the class of demicontractive operators, including the classes of quasi-nonexpansive and strictly pseudocontractive operators. The considered algorithm is written as: x _n+1 := (1 − w) v _n + w T v _n , v _n := x _n − α _n Θ′(x _n ), where w ∈ (0,1) and , Θ′ is the Gateaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x _n ) generated, with an arbitrary , by this scheme converges strongly to some element in Argmin _Fix(T) Θ.      

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.     

The minimality of the map \frac{x}{\|{x}\|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ u‖^p dx, where f is a non-negative function. We prove that in each of the two following cases:

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

The coarse Lipschitz geometry of \ell_p\oplus\ell_q

We show that for 1 < p < ∞ with p ≠ 2 the space L _p (0,1) is not uniformly homeomorphic to . We also show that if 1 < p < 2 < q < ∞ the space has unique uniform structure, answering a question of Johnson, Lindenstrauss and Schechtman (Geom. Funct. Anal. 6:430–470, 1996). The first author was supported by NSF grant DMS-0555670 and the second author was supported by NSF grant DMS-0701097.     

Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Δ u=u^p in Ω, u=0 on ∂Ω, where Ω is a bounded domain of , n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and , as a consequence of the Pohožaev's identity). Mathematics Subject Classification (2000) 35J20, 35J60, 35J65     

A blowing-up branch of solutions for a mean field equation

We consider the equation

Asymptotic Behaviour of Iterates of Volterra Operators on L p (0, 1)

Given k ∈ L¹ (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V_k defined on L^p (0,1) by