Solution of boundary value problems for the Laplace equation in a ball bounded by a multilayer film

We derive boundary conditions on multilayer films bounding a ball and consisting of infinitely thin strongly and weakly permeable layers and obtain formulas expressing the solutions of boundary value problems for the Laplace equation in a ball bounded with two-layer films by single quadratures via the solutions of the classical Dirichlet and Neumann problems for the Laplace equation in the ball (without the films).     

Blow up and grazing collision in viscous fluid solid interaction systems

We investigate qualitative properties of strong solutions to a classical system describing the fall of a rigid ball under the action of gravity inside a bounded cavity filled with a viscous incompressible fluid. We prove contact between the ball and the boundary of the cavity implies blow up of strong solutions and such a contact has to occur in finite time under symmetry assumptions on the initial data.     

On exterior Neumann problems with an asymptotically linear nonlinearity

In this paper, we study exterior Neumann problems with an asymptotically linear nonlinearity. We establish the existence of ground state solutions. Furthermore when the domain is a complement of a ball, we prove the ground state solutions are not radially symmetric. We also give asymptotic profiles of ground state solutions.     

The problem of optimal control of a Chaplygin ball by internal rotors

We study the problem of optimal control of a Chaplygin ball on a plane by means of 3 internal rotors. Using Pontryagin maximum principle, the equations of extremals are reduced to Hamiltonian equations in group variables. For a spherically symmetric ball, the solutions can be expressed in by elliptic functions.     

Symmetry-breaking at positive solutions

We consider symmetry-breaking bifurcations at positive solutions of semilinear elliptic boundary value problems in a ball, with Dirichlet boundary conditions. We prove that symmetry-breaking can only occur by bifurcation of axisymmetric solutions. We also obtain sufficient conditions for such symmetry-breaking.Dedicated to Professor H. Knobloch at the occasion of his 60th birthday     

Ricci Flow of Regions with Curvature Bounded Below in Dimension Three

We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming an open ball at time zero of radius one has sectional curvature bounded from below by ?1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well-defined time interval. The estimates we obtain depend only on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. Versions of these estimates for balls of radius r follow using scaling arguments.     

Generalized third boundary value problem for the biharmonic equation

We study the existence and uniqueness of solutions of a generalized third boundary value problem for the inhomogeneous biharmonic equation in the unit ball.     

New concentration phenomena for a class of radial fully nonlinear equations

We study radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems in a ball, driven by the extremal Pucci's operators and with a power nonlinear term. We first determine a new critical exponent related to the existence or nonexistence of such solutions. Then we analyze the asymptotic behavior of the radial nodal solutions as the exponents approach the critical values, showing that new concentration phenomena occur. Finally we define a suitable weighted energy for these solutions and compute its limit value.     

Some remarks on singular solutions of nonlinear elliptic equations. I

We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes. Dedicated to Felix Browder on his 80th birthday     

Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n?3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n?4, we give conditions on the function h to guarantee there is only one simple blow-up point.     

On the spatial distribution of solutions of decomposable form equations

We study the distribution in space of the integral solutions to an integral decomposable form equation, by considering the images of these solutions under central projection onto a unit ball. If we think of the solutions as stars in the night sky, we ask what constellations are visible from the earth (the unit ball). Answers are given for a large class of examples which are then illustrated using the software packages KANT and Maple. These pictures highlight the accuracy of our predictions and arouse interest in cases not covered by our results. Within the range of applicability of our results lie solutions to norm form equations and units in abelian group rings. Thus our theory has a lot to say about where these interesting objects can be found and what they look like.

     

Quelques Inequations Quasi-Variationnelles avec Restrictions Bornees par Rapport a l'Ordre

We establish sufficient conditions for the existence of solutions of some quasi-variational inequalities with non-monotone restriction, but which is order bounded on a ball with sufficiently large radius.     

四元数分析中超球与双圆柱区域上的正则函数

本文讨论了四元数分析中的正则函数Ｕ（ｚ）（满足方程ｚＵ（ｚ）＝０,ｚ＝ｘ１＋ｉｘ２＋ｊｘ３－ｋｘ４）及其边值问题,给出了超球与双圆柱区域上的四元数正则函数的Ｃａｕｃｈｙ积分公式,获得了一般区域上正则函数的无穷次可微性;给出了定义在超球与双圆柱区域边界上的四元数函数可正则开拓到区域内的条件;讨论了满足非齐次方程ｚＦ＝ｆ的四元函数Ｆ（ｚ）的Ｄｉｒｉｃｈｌｅｔ和Ｎｅｕｍａｎｎ边值问题;获得了超球与双圆柱区域上这两种边值问题解的积分表示．     

On the absence of global solutions to the gauss equation and solutions in external areas

We consider conditions under which the Gauss equation has no solutions defined in the whole space or in areas external with respect to a ball. The absence of solutions in external areas is established in the case when the number of independent variables is more than two. In the two-dimensional case we obtain conditions ensuring the absence of global solutions to the second-order elliptic equation with variable coefficients in its linear part.     

How to control the Chaplygin ball using rotors. II

In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.     

Spherically symmetric internal layers for activator-inhibitor systems II. Stability and symmetry breaking bifurcations

A reaction-diffusion system of activator-inhibitor type is studied on an N-dimensional ball with the homogeneous Neumann boundary conditions. We analyze the stability property of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions which are not spherically symmetric.     

Nonnegative solutions to a semilinear dirichlet problem in a ball are positive and radially syhhetric*

We prove that nonnegative.so1utions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the .domain which ensures that nonnegative solutions are positive in the interior.     

Existence and uniqueness of positive radial solutions for the Lane–Emden system

In this article, we consider (component-wise) positive radial solutions of a weakly coupled system of elliptic equations in a ball with homogeneous nonlinearities. The existence is well-known in general: We give a result for the remaining cases. The uniqueness is less studied: We complement the known results.     

Dirichlet problem in the ball for an equation with a delta-like coefficient

We study the Dirichlet problem in the ball for the Helmholtz equation with an additional term in the form of the product by the delta function. The additional term is approximated by the simplest integral expression, and the solution of the original equation is defined as the limit of solutions of the regularized equations. We obtain exact solutions depending on the chosen method of approximation.     

Examples of stationary Euler flows

We construct concrete solutions of the incompressible Euler equation in a domain of the cylinder type, and the compressible Euler equation outside a ball (or a rotationary symmetric body) with a constant speed at infinity along the axis of symmetry.