相关论文: On the trajectories of O(1)-Kepler Problems

In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are invariant…

The non-transversal intersection of the free boundary with the fixed boundary is obtained for nonlinear uniformly elliptic operators when $\Omega = \{\nabla u \neq 0\} \cap \{x_n>0\}$ thereby solving a problem in elliptic theory that in the…

The motion of a charged particle in a nonuniform straight magnetic field with a uniform magnetic-field gradient is solved exactly in terms of elliptic functions. The connection between this problem and the guiding-center approximation is…

The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have…

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class $[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

The motion of binary star systems is re-examined in the presence of perturbations from the theory of general relativity. The Kepler problem is regularized and linearized with quaternions. In this way first order perturbation results are…

We provide a geometric optics description in spaces of low regularity, $L^2$ and $H^1$, of the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze the boundary…

Transition to a nonrelativistic Pauli equation in Riemann space of constant positive curvature for a Dirac particle in presence of the Coulomb field is performed in the system of radial equations, exact solutions are constructed in terms of…

We consider the positive solutions of the nonlinear eigenvalue problem $-\Delta_{\mathbb{H}^n} u = \lambda u + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(\Omega),$ where $\Omega$ is a geodesic ball of radius $\theta_1$ on…

In this paper, we address the numerical solution of the Optimal Transport Problem on undirected weighted graphs, taking the shortest path distance as transport cost. The optimal solution is obtained from the long-time limit of the gradient…

We study the inverse problem of determining the coefficients of the fractional power of a general second order elliptic operator given in the exterior of an open subset of the Euclidean space. We show the problem can be reduced into…

Mechanics texts tell us that a particle in a bound orbit under gravitational central force moves on an ellipse, while introductory physics texts approximate the earth as flat, and tell us that the particle moves in a parabola. The…

For each simple euclidean Jordan algebra $V$ of rank $\rho$ and degree $\delta$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary…

In a Keplerian system, a large number of bodies orbit a central mass. Accretion disks, protoplanetary disks, asteroid belts, and planetary rings are examples. Simulations of these systems require algorithms that are computationally…

We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of…

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $\mathbb{R}^n$ \[ (-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1} \] in the convex case $1\leq q<2_s^*-1$, where $…

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator…

We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of…

Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} \bigl(-\frac{\mathrm{d}^2}{\mathrm{d} x^2}\bigr)^m u + \lambda G'(u) = F'(u)\\ \int_{\mathbb{R}} K(u) \,…