相关论文: Block regularization of the Kepler problem on surf…
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide…
We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature…
We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole…
The global symmetry transformations generated by Runge-Lenz vector of twodimensional Kepler problem are explicitly described. They are given in terms of SU(2) left group multiplication with group elements being suitably parametrized by…
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary…
A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open…
In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…
We treat the circular and elliptic restricted three-body problems in inertial frames as periodically forced Kepler problems with additional singularities and explain that in this setting the main result of [4] is applicable. This guarantees…
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for…
We introduce a novel regularization framework for the two-dimensional incompressible Euler equation that exactly preserves the transport structure of multi-phase vorticity fields. The key step is a reformulation of multi-phase vortex patch…
We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation…
The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the…
The generalization of Bertrand's theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analyzed for the case of…
We consider the Cahn-Hilliard equation with constant mobility and logarithmic potential on a two-dimensional evolving closed surface embedded in $\mathbb R^3$, as well as a related weighted model. The well-posedness of weak solutions for…
For a local suitable weak solution to the Navier-Stokes equations, we prove that if the vorticity vectors belong to a double cone in regions of high vorticity magnitude, then the solution is regular. Roughly speaking this implies that, near…
We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described…
The logarithm function is the gravitational potential in $\mathbb{R}^2$. We prove that the logarithm central force problem is block regularizable, that is, the (incomplete) flow may be continuously extended over the singularity at the…
Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler…