Related papers: Minimum Energy Problem on the Hypersphere
It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we…
As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi/c$,…
We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular…
We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion…
As shown by Overhauser and others, accurate pair densities for the uniform electron gas may be found by solving a two-electron scattering problem with an effective screened electron-electron repulsion. In this work we explore the extension…
We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-\Delta)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $\Omega,$ } { \mathcal{N}_{s}u=0 }…
Minimal dark energy models, described by the same number of free parameters of the standard cosmological model with cold dark matter plus a cosmological constant to parameterize the dark energy component, constitute very appealing scenarios…
We consider the problem of allocating a finite number of heat sources in the n-dimensional sphere. When only one such source -assumed to be of infinite temperature- is placed and assuming a constant cooling rate in the sphere, we prove that…
Let $B^n\subset {\mathbb R}^{n}$ and ${\mathbb S}^n\subset {\mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left…
A problem of self-energy and self-force for a charged point-like particle in a higher dimensional homogeneous gravitational field is considered. We study two cases, when a particle has usual electric charge and a case when it has a scalar…
In models of $N$ interacting particles in $\R^d$ as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function $c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big)$ where $\ell: \R_+ \to [0,\infty]$…
Equation of motion of Sommerfeld sphere in the one-dimensional potential hole, produced by two equal charges on some distance from each other, is numerically investigated. Two types of solutions are found: (i) damping oscillations, (ii)…
In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data…
We use the isotropy of the Cosmic Microwave Background to place stringent constraints on a possible electrical charge asymmetry of the universe. We find the excess charge per baryon to be $q_{e-p}<10^{-26}e$ in the case of a uniform…
We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of…
We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence…
We discuss the possibility of a laboratory search for light minicharged particles carrying electric charge that is a small fraction $\epsilon$ of that of electron. We point out that the production of pairs of such particles in an electric…
We consider two point charges in electrostatic interaction between them within the framework of a nonlinear model, associated with QED, that provides finiteness of their field energy. We argue that if the two charges are equal to each other…
We investigate bound state solutions of the 2D Schr\"odinger equation with a dipole potential originating from the elastic effects of a single edge dislocation. The knowledge of these states could be useful for understanding a wide variety…
We consider the van der Waals energy of an atom near the infinitely thin sphere with finite conductivity which model the fullerene. We put the sphere into spherical cavity inside the infinite dielectric media, then calculate the energy of…