Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system of integral equations

In this article, we study the asymptotics of the positive solutions of the Euler–Lagrange system of the weighted Hardy–Littlewood–Sobolev in R ⁿ $$\begin{array}{ll} u(x) = \frac{1}{|x|^{\alpha}}\int\limits_{R^{n}} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int\limits_{R^{n}} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}} dy.\end{array}$$ A new iterative method is introduced to obtain the optimal weighted local integrability of u(x). By this new method, we establish the asymptotic estimates of the solutions around the origin and near infinity. With these new estimates, we complete the study of the asymptotic behavior of the solutions. We believe this new iterative method and the new type of the weighted local estimates can be used in many other cases.     

Hardy–Littlewood–Sobolev inequality on the parabolic biangle

The Ramanujan Journal - We define the heat semigroup associated with a system of bivariate Jacobi polynomials which are orthogonal with respect to a probability measure on the parabolic biangle...     

Symmetry and regularity of extremals of an integral equation related to the Hardy�CSobolev inequality

Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method.     

On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality

We show existence of minimizers for the Hardy–Sobolev–Maz’ya inequality in when either m > 2, n≥ 1 or m = 1, n≥ 3. The authors expresses their gratitude to the faculties of mathematics departments at Technion - Haifa Institute of Technology, at the University of Crete and at the University of Cyprus for their hospitality. A.T. acknowledges partial support by the RTN European network Fronts–Singularities, HPRN-CT-2002-00274. K.T acknowledges support as a Lady Davis Visiting Professor at Technion and partial support from University of Crete, University of Cyprus and Swedish Research Council.     

Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

Regularity properties of the solutions to the 3D Maxwell–Landau–Lifshitz system in weighted Sobolev spaces

This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of _∂tm,_∂tE${\mathrm{\partial }}_t\mathbf{m},{\mathrm{\partial }}_t\mathbf{E}$ and _∂tH${\mathrm{\partial }}_t\mathbf{H}$ in the spaces L^∞(I,L²(Ω))$L^{\infty }(I,L^2(\Omega ))$ and L²(I,W^1,2(Ω))$L^2(I,W^{1,2}(\Omega ))$ we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme.     

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u|$\left|u\right|$ weighted HLS inequality in Theorem 1.1 and the |z|$\left|z\right|$ weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t)$u=(z,t)$ as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u|$\left|u\right|$ weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.     

A functional framework for the Keller–Segel system: Logarithmic Hardy–Littlewood–Sobolev and related spectral gap inequalities

This Note is devoted to several inequalities deduced from a special form of the logarithmic Hardy–Littlewood–Sobolev, which is well adapted to the characterization of stationary solutions of a Keller–Segel system written in self-similar variables, in case of a subcritical mass. For the corresponding evolution problem, such functional inequalities play an important role for identifying the rate of convergence of the solutions towards the stationary solution with same mass.     

On the necessity of the assumptions used to prove Hardy–Littlewood and Riesz rearrangement inequalities

We prove that supermodularity is a necessary condition for the generalized Hardy–Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz-type integral.     

Asymptotic approximations to the Hardy–Littlewood function

The function Q(x):=_∑n≥1(1/n)sin(x/n)$Q\left(x\right):={\sum }_{n\ge 1}(1/n)sin(x/n)$ was introduced by Hardy and Littlewood (1936) [5] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos (2005) [3] of a conjecture by Clark and Ismail (2003) [14]. More precisely, Alzer et al. have shown that the Clark and Ismail conjecture is true if and only if Q(x)≥−π/2$Q\left(x\right)\ge -\pi /2$ for all x>0$x>0$. It is known that Q(x)$Q\left(x\right)$ is unbounded in the domain x∈(0,∞)$x\in (0,\infty )$ from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point x$x$ for which Q(x)<−π/2$Q\left(x\right)<-\pi /2$. This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate Q(x)$Q\left(x\right)$ for very large values of x$x$. In this paper we continue the work started by Gautschi (2005) in [4] and develop several approximations to Q(x)$Q\left(x\right)$ for large values of x$x$. We use these approximations to find an explicit value of x$x$ for which Q(x)<−π/2$Q\left(x\right)<-\pi /2$.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.     

On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 99–115, January, 2009.     

Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).