1,419 research outputs found
Convex Tours of Bounded Curvature
We consider the motion planning problem for a point constrained to move along
a smooth closed convex path of bounded curvature. The workspace of the moving
point is bounded by a convex polygon with m vertices, containing an obstacle in
a form of a simple polygon with vertices. We present an O(m+n) time
algorithm finding the path, going around the obstacle, whose curvature is the
smallest possible.Comment: 11 pages, 5 figures, abstract presented at European Symposium on
Algorithms 199
Eigenvalue upper bounds for the magnetic Schroedinger operator
We study the eigenvalues of the magnetic Schroedinger operator associated
with a magnetic potential A and a scalar potential q, on a compact Riemannian
manifold M, with Neumann boundary conditions if the boundary is not empty. We
obtain several bounds for the spectrum. Besides the dimension and the volume of
the manifold, the geometric quantity which plays an important role in these
estimates is the first eigenvalue of the Hodge-de Rham Laplacian acting on
co-exact 1-forms. In the 2-dimensional case, this is nothing but the first
positive eigenvalue of the Laplacian acting on functions. As for the dependence
of the bounds on the potentials, it brings into play the mean value of the
scalar potential q, the L^2-norm of the magnetic field B=dA, and the distance,
taken in L^2, between the harmonic component of A and the subspace of all
closed 1-forms whose cohomology class is integral (that is, having integral
flux around any loop). In particular, this distance is zero when the first
cohomology group is trivial.Comment: This preprint partially replaces arXiv: 1611.0193
Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations
In this article, we provide existence results for a general class of nonlocal
and nonlinear second-order parabolic equations. The main motivation comes from
front propagation theory in the cases when the normal velocity depends on the
moving front in a nonlocal way. Among applications, we present level-set
equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo
systems
On "many black hole" space-times
We analyze the horizon structure of families of space times obtained by
evolving initial data sets containing apparent horizons with several connected
components. We show that under certain smallness conditions the outermost
apparent horizons will also have several connected components. We further show
that, again under a smallness condition, the maximal globally hyperbolic
development of the many black hole initial data constructed by Chrusciel and
Delay, or of hyperboloidal data of Isenberg, Mazzeo and Pollack, will have an
event horizon, the intersection of which with the initial data hypersurface is
not connected. This justifies the "many black hole" character of those
space-times.Comment: several graphic file
On the placement of an obstacle so as to optimize the Dirichlet heat trace
We prove that among all doubly connected domains of bounded by two
spheres of given radii, , the trace of the heat kernel with Dirichlet
boundary conditions, achieves its minimum when the spheres are concentric
(i.e., for the spherical shell). The supremum is attained when the interior
sphere is in contact with the outer sphere.This is shown to be a special case
of a more general theorem characterizing the optimal placement of a spherical
obstacle inside a convex domain so as to maximize or minimize the trace of the
Dirichlet heat kernel. In this case the minimizing position of the center of
the obstacle belongs to the "heart" of the domain, while the maximizing
situation occurs either in the interior of the heart or at a point where the
obstacle is in contact with the outer boundary. Similar statements hold for the
optimal positions of the obstaclefor any spectral property that can be obtained
as a positivity-preserving or positivity-reversing transform of
,including the spectral zeta function and, through it, the regularized
determinant.Comment: in SIAM Journal on Mathematical Analysis, Society for Industrial and
Applied Mathematics, 201
Local and global properties of solutions of quasilinear Hamilton-Jacobi equations
We study some properties of the solutions of (E) \;-\Gd_p u+|\nabla u|^q=0
in a domain \Gw \sbs \BBR^N, mostly when . We give a universal
priori estimate of the gradient of the solutions with respect to the distance
to the boundary. We give a full classification of the isolated singularities of
the positive solutions of (E), a partial classification of isolated
singularities of the negative solutions. We prove a general removability result
in expressed in terms of some Bessel capacity of the removable set. We extend
our estimates to equations on complete non compact manifolds satisfying a lower
bound estimate on the Ricci curvature, and derive some Liouville type theorems.Comment: to appear J. Funct. Ana
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