Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems

Let B=B₁(0) be the unit ball in Rn and r=|x|. We study the poly-harmonic Dirichlet problem

     

一类积分方程组正解的对称性和单调性

本文讨论积分方程组（？）解的性质,其中G_α是α阶贝塞尔位势核,0≤β〈α（n-α＋β）/n,1/（q＋1）＋1/（r＋1）〉（n-α＋β）/n,1/（r＋1）＋1/（p＋1）〉（n-α＋β）/n.我们用积分形式的移动平面法证明上述积分方程组的正解是径向对称且单调的.     

Quantitative analysis of some system of integral equations

In this paper, we study the integrability of the non-negative solutions to the Euler-Lagrange equations associated with Weighted Hardy-Littlewood-Sobolev (HLS) inequality. We obtain the optimal integrability for the solutions. The integrability and the radial symmetry (which we derived in our earlier paper) are the key ingredients to study the growth rate at the center and the decay rate at infinity of the solutions. These are also the essential properties needed to classify all non-negative solutions. Some simple generalizations are also provided here.     

Radial solutions of an elliptic equation with singular nonlinearity

For the equation

     

Optimal martingale transport between radially symmetric marginals in general dimensions

We determine the optimal structure of couplings for the Martingale transport problem between radially symmetric initial and terminal laws $\mu ,\nu$ on ${\mathbb{R}}^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\mathbb{E}f(\left|\right|X-Y\left|\right|)$ where $f$ is concave and strictly increasing, and the dimension $d$ is arbitrary.     

带权函数的积分方程组正解的对称性和单调性

主要研究全空间上一类带权函数的积分方程组正解的径向对称性和单调性问题．在合适条件下,主要利用积分形式的移动平面方法,Hardy—Littlewwood—Sobolev（HLS）和Holder不等式给出了积分方程组正解的径向对称性和单调性的结论．这一结论很好的推广了已有的结果．     

Nonclassical symmetry analysis for hyperbolic partial differential equation

We consider the nonclassical symmetry of one-dimensional hyperbolic differential equations of the form u_t + M(u)u_x = 0. For the infinitesimal generator $\mathbf{V}=\tau {\mathrm{\partial }}_t+\xi {\mathrm{\partial }}_x+{\sum }_{i=1}^n{\varphi }_i{\mathrm{\partial }}_{u_i}$, it is shown that ξ is an eigenvalue of the matrix M when ϕ_i = 0 [Souichi M. Nonclassical symmetry and Riemann invariants. Int J Nonlinear Mech, [in press]]. In this paper, we prove a sufficient condition of a lemma.     

A local monotonicity formula for an inhomogenous singular perturbation problem and applications

In this paper we prove a local monotonicity formula for solutions to an inhomogeneous singularly perturbed diffusion problem of interest in combustion. This type of monotonicity formula has proved to be very useful for the study of the regularity of limits u of solutions of the singular perturbation problem and of ∂{u > 0}, in the global homogeneous case. As a consequence of this formula we prove that u has an asymptotic development at every point in ∂{u > 0} where there is a nonhorizontal tangent ball. These kind of developments have been essential for the proof of the regularity of ∂{u > 0} for Bernoulli and Stefan free boundary problems. We also present applications of our results to the study of the regularity of ∂{u > 0} in the stationary case including, in particular, its regularity in the case of energy minimizers. We present as well a regularity result for traveling waves of a combustion model that relies on our monotonicity formula and its consequences.The fact that our results hold for the inhomogeneous problem allows a very wide applicability. Indeed, they may be applied to problems with nonlocal diffusion and/or transport. The research of the authors was partially supported by Fundación Antorchas Project 13900-5, Universidad de Buenos Aires grant X052, ANPCyT PICT No 03-13719, CONICET PIP 5478. The authors are members of CONICET.     

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the (3+1)-dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov-Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases , the constant mass and the non-relativistic limits are briefly investigated.     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

The relation between non-bossiness and monotonicity

It is a well-known fact that in some economic environments, non-bossiness and monotonicity are interrelated. In this paper, we have presented a new domain-richness condition called weak monotonic closedness, on which non-bossiness in conjunction with individual monotonicity is equivalent to monotonicity. Moreover, by applying our main result to several types of economies, we have obtained characterizations in terms of non-bossiness.     

Radial symmetry and uniqueness for an overdetermined problem

Consider a function u, harmonic in a ring‐shaped domain and taking two constant (distinct) values on the two connected components of the boundary. If we know in advance that one of the components is a sphere, and that u satisfies some overdetermined condition on the other one, can we conclude that u is radial? This paper answers this question for certain overdetermined conditions on the gradient of u, generalizing some previous results. Conditions depending on the principal curvatures of the boundary are also investigated. Existence and uniqueness of a radial solution to the overdetermined problem are discussed. Some extensions to ellipsoidal domains, as well as to quasilinear elliptic equations, are carried out. Copyright © 2001 John Wiley & Sons, Ltd.     

Mountain Pass solutions for quasi-linear equations via a monotonicity trick

We obtain the existence of symmetric Mountain Pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity trick proved in Squassina (in press) [22]. The main results are new also for the p-Laplacian operator.