235 research outputs found
Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound
We complete the picture of sharp eigenvalue estimates for the p-Laplacian on
a compact manifold by providing sharp estimates on the first nonzero eigenvalue
of the nonlinear operator when the Ricci curvature is bounded from
below by a negative constant. We assume that the boundary of the manifold is
convex, and put Neumann boundary conditions on it. The proof is based on a
refined gradient comparison technique and a careful analysis of the underlying
model spaces.Comment: Sign mistake fixed in the proof of the gradient comparison theorem
(theorem 5.1 pag 10), and some minor improvements aroun
Z_2-Bi-Gradings, Majorana Modules and the Standard Model Action
The action functional of the Standard Model of particle physics is intimately
related to a specific class of first order differential operators called Dirac
operators of Pauli type ("Pauli-Dirac operators"). The aim of this article is
to carefully analyze the geometrical structure of this class of Dirac operators
on the basis of real Dirac operators of simple type. On the basis of simple
type Dirac operators, it is shown how the Standard Model action (STM action)
may be viewed as generalizing the Einstein-Hilbert action in a similar way the
Einstein-Hilbert action is generalized by a cosmological constant. Furthermore,
we demonstrate how the geometrical scheme presented allows to naturally
incorporate also Majorana mass terms within the Standard Model. For reasons of
consistency these Majorana mass terms are shown to dynamically contribute to
the Einstein-Hilbert action by a "true" cosmological constant. Due to its
specific form, this cosmological constant can be very small. Nonetheless, this
cosmological constant may provide a significant contribution to dark
matter/energy. In the geometrical description presented this possibility arises
from a subtle interplay between Dirac and Majorana masses
Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians
We consider the problem of minimising the th eigenvalue, , of
the (-)Laplacian with Robin boundary conditions with respect to all domains
in of given volume . When , we prove that the second
eigenvalue of the -Laplacian is minimised by the domain consisting of the
disjoint union of two balls of equal volume, and that this is the unique domain
with this property. For and , we prove that in many cases a
minimiser cannot be independent of the value of the constant in the
boundary condition, or equivalently of the volume . We obtain similar
results for the Laplacian with generalised Wentzell boundary conditions .Comment: 16 page
The heart of a convex body
We investigate some basic properties of the {\it heart}
of a convex set It is a subset of
whose definition is based on mirror reflections of euclidean
space, and is a non-local object. The main motivation of our interest for
is that this gives an estimate of the location of the
hot spot in a convex heat conductor with boundary temperature grounded at zero.
Here, we investigate on the relation between and the
mirror symmetries of we show that
contains many (geometrically and phisically) relevant points of
we prove a simple geometrical lower estimate for the diameter of
we also prove an upper estimate for the area of
when is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic
and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6
A remark on an overdetermined problem in Riemannian Geometry
Let be a Riemannian manifold with a distinguished point and
assume that the geodesic distance from is an isoparametric function.
Let be a bounded domain, with , and consider
the problem in with on ,
where is the -Laplacian of . We prove that if the normal
derivative of along the boundary of is a
function of satisfying suitable conditions, then must be a
geodesic ball. In particular, our result applies to open balls of
equipped with a rotationally symmetric metric of the form
, where is the standard metric of the sphere.Comment: 8 pages. This paper has been written for possible publication in a
special volume dedicated to the conference "Geometric Properties for
Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in
Palinuro in May 201
Second-order -regularity in nonlinear elliptic problems
A second-order regularity theory is developed for solutions to a class of
quasilinear elliptic equations in divergence form, including the -Laplace
equation, with merely square-integrable right-hand side. Our results amount to
the existence and square integrability of the weak derivatives of the nonlinear
expression of the gradient under the divergence operator. This provides a
nonlinear counterpart of the classical -coercivity theory for linear
problems, which is missing in the existing literature. Both local and global
estimates are established. The latter apply to solutions to either Dirichlet or
Neumann boundary value problems. Minimal regularity on the boundary of the
domain is required. If the domain is convex, no regularity of its boundary is
needed at all
On the regularity up to the boundary for certain nonlinear elliptic systems
We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems
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