22,252 research outputs found
A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem
After the existence proof of the first remarkably stable simple choreographic
motion-- the figure eight of the planar three-body problem by Chenciner and
Montgomery in 2000, a great number of simple choreographic solutions have been
discovered numerically but very few of them have rigorous existence proofs and
none of them are stable. Most important to astronomy are stable periodic
solutions which might actually be seen in some stellar system. A question for
simple choreographic solutions on -body problems naturally arises: Are there
any other stable simple choreographic solutions except the figure eight?
In this paper, we prove the existence of infinitely many simple choreographic
solutions in the classical Newtonian 4-body problem by developing a new
variational method with structural prescribed boundary conditions (SPBC).
Surprisingly, a family of choreographic orbits of this type are all linearly
stable. Among the many stable simple choreographic orbits, the most
extraordinary one is the stable star pentagon choreographic solution. The star
pentagon is assembled out of four pieces of curves which are obtained by
minimizing the Lagrangian action functional over the SPBC.
We also prove the existence of infinitely many double choreographic periodic
solutions, infinitely many non-choreographic periodic solutions and uncountably
many quasi-periodic solutions. Each type of periodic solutions have many stable
solutions and possibly infinitely many stable solutions.Comment: Total 26 pages including 5 pages of figures and references, 12
figure
The Dirichlet Problem for Curvature Equations in Riemannian Manifolds
We prove the existence of classical solutions to the Dirichlet problem for a
class of fully nonlinear elliptic equations of curvature type on Riemannian
manifolds. We also derive new second derivative boundary estimates which allows
us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck
[4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions
and less convex domains.Comment: 32 pages, no figures. Final version. Paper accepted to publication in
Indiana University Mathematics Journa
Initial boundary value problems for Einstein's field equations and geometric uniqueness
While there exist now formulations of initial boundary value problems for
Einstein's field equations which are well posed and preserve constraints and
gauge conditions, the question of geometric uniqueness remains unresolved. For
two different approaches we discuss how this difficulty arises under general
assumptions. So far it is not known whether it can be overcome without imposing
conditions on the geometry of the boundary. We point out a natural and
important class of initial boundary value problems which may offer
possibilities to arrive at a fully covariant formulation.Comment: 19 page
Starshaped locally convex hypersurfaces with prescribed curvature and boundary
In this paper we find strictly locally convex hypersurfaces in
with prescribed curvature and boundary. The main result is
that if the given data admits a strictly locally convex radial graph as a
subsolution, we can find a radial graph realizing the prescribed curvature and
boundary. As an application we show any smooth domain on the boundary of a
compact strictly convex body can be deformed to a smooth hypersurface with the
same boundary (inside the convex body) and realizing any prescribed curvature
function smaller than the curvature of the body.Comment: 24 pages. References updated. Published in The Journal of Geometric
Analysi
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