Gradient estimates for p(x)-harmonic functions

We establish an a priori gradient estimate for the solutions of the quasilinear p(x)-Laplace equation Δ _p(x) u = 0.     

The boundary Harnack inequality for infinity harmonic functions in the plane

We prove the boundary Harnack inequality for positive infinity harmonic functions vanishing on a portion of the boundary of a bounded domain under the assumption that is a quasicircle.

     

Local {W^{2,m(\cdot)}_{loc}}regularity for p(.)-Laplace equations

We study ${W^{2,m(\cdot)}_{loc}}$ regularity for local weak solutions of p(·)-Laplace equations where ${p\in C^1(\Omega) \cap C(\overline{\Omega})}$ and ${\min_{x\in \overline{\Omega}} p(x) > 1}$ .     

The boundary Harnack inequality for infinity harmonic functions in Lipschitz domains satisfying the interior ball condition

In this note, we give a short proof for the boundary Harnack inequality for infinity harmonic functions in a Lipschitz domain satisfying the interior ball condition. Our argument relies on the use of quasiminima and the notion of comparison with cones.     

A note about the generalized Hardy-Sobolev inequality with potential in L^{{p,d}} {\left( {\mathbb{R}^{n} } \right)}

We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential is replaced by any potential V belonging to the Lorentz space . We show that the best constant in these inequalities is achieved provided that where . We also analyze the limit case . Finally an application to a non-linear eigenvalues problem with rough potentials is presented.Received: 19 September 2004, Accepted: 15 November 2004, Published online: 22 December 2004     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

Monotone convergence of iterative methods for (n,p) and (p,n) boundary value problems

We provide sufficient conditions so that the Picard iterative scheme for (n, p) and (p, n) boundary value problems converges monotonically. An example which illustrates the usefulness of this technique is also included.     

L p estimates for the iterated Hardy-Littlewood maximal operator on\mathbb{R}^n and K n , k a local fieldand K n , k a local field

In light of two measure estimate inequalities from [4] for the iterated Hardy-Littlewood maximal operatorM ^k f, certain equivalence betweenM ^k f and the Zygmund classLLog ^a L are established on $\mathbb{R}^n $ , so that we generalize Stein'sLLogL theorem. In Section 3, a simple induction enables us to prove such extensions onK ⁿ, the n-dimensional linear space over a local fieldK, without recoursing to Leckband's result.     

The theorem on convergence with a functional for integral functionals with p(x)- and p(x, u)-growth

We continue studying weak convergence for the integral functionals satisfying p(x)- and p(x, u)-growth conditions. We obtain the theorem on convergence with a functional and some results on the relation between integral functionals and their abstract lower semicontinuous extensions.     

J-inner matrix functions, interpolation and inverse problems for canonical systems, V: The inverse input scattering problem for Wiener class and rational p×q input scattering matrices

The general formulas developed in the fourth paper in this series are applied to solve the inverse input scattering problem for canonical integral systems in the special cases that the input scattering matrix is ap×q matrix valued function in the Wiener class (and the associated pairs are homogeneous). These formulas are then further specialized to the rational case. Whenp=q, these formulas are connected to the earlier results of Alpay-Gohberg and Gohberg-Kaashoek-Sakhnovich, who studied inverse problems for a related system of differential equations.This research was partially supported by a Minerva Foundation grant that is acknowledged with thanks.     

The green function for the weighted biharmonic operator Δ(1−∣z∣2)−αΔ, and factorization of analytic functions, and factorization of analytic functions

An explicit integral formula is obtained for the Green function of the weighted biharmonic operator Δ(1−∣z∣²)^−αΔ in the unit disk of the complex plane for the case α ∈ (−1, 0). The formula shows the positivity of the Green function. This is a basis for a theorem on factorization of analytic functions in the weighted Bergman spaces with the weights ω(z)=(1−∣z∣²)^α as products of a nonvanishing function and a function of special form responsible for the zeros. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 203–221.     

On the completeness of the functions e−np(x) cos nx,e−np(x) sin nx for n ≥ 0 and p(x) a 2π periodic function

It is shown how a conjecture of G. B. Whitham on the completeness of the functions { ψ _n } = { e^?np(x) cos nx, e^?np(x) sin nx } in the interval [0, 2π] is indeed true provided p has a Hölder continuous first derivative, i.e., p ∈ C^{1, α}[0, 2π]. Also an elementary proof of Whitham's conjecture is given for p ∈ C^{2, α}[0, 2π].     

Operator and matrix representation for the generalized inverse AT,S(2)A_{T,S}^{(2)}

This paper presents the matrix representation for extension of inverse of restriction of a linear operator to a subspace, on the basis of which we establish useful representations in operator and matrix form for the generalized inverse A_T,S⁽²⁾A_{T,S}^{(2)} and give some of their applications.