174 research outputs found
The CR Paneitz Operator and the Stability of CR Pluriharmonic Functions
We give a condition which ensures that the Paneitz operator of an embedded
three-dimensional CR manifold is nonnegative and has kernel consisting only of
the CR pluriharmonic functions. Our condition requires uniform positivity of
the Webster scalar curvature and the stability of the CR pluriharmonic
functions for a real analytic deformation. As an application, we show that the
real ellipsoids in are such that the CR Paneitz operator is
nonnegative with kernel consisting only of the CR pluriharmonic functions.Comment: 11 pages; final versio
Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
We consider the following eigenvalue optimization problem: Given a bounded
domain and numbers , ,
find a subset of area for which the first Dirichlet
eigenvalue of the operator is as small as possible.
We prove existence of solutions and investigate their qualitative properties.
For example, we show that for some symmetric domains (thin annuli and dumbbells
with narrow handle) optimal solutions must possess fewer symmetries than
; on the other hand, for convex reflection symmetries are
preserved.
Also, we present numerical results and formulate some conjectures suggested
by them.Comment: 24 pages; 3 figures (as separate files); (shortened previous
version); to appear in Comm. Math. Phy
Local Asymmetry and the Inner Radius of Nodal Domains
Let M be a closed Riemannian manifold of dimension n. Let f be an
eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue
\lambda. We show that the volume of {f>0} inside any ball B whose center lies
on {f=0} is > C|B|/\lambda^n. We apply this result to prove that each nodal
domain contains a ball of radius > C/\lambda^n.Comment: 12 pages, 1 figure; minor corrections; to appear in Comm. PDE
A Remark on the Geometry of Uniformly Rotating Stars
In this paper we classify the free boundary associated to equilibrium
configurations of compressible, self-gravitating fluid masses, rotating with
constant angular velocity. The equilibrium configurations are all critical
points of an associated functional and not necessarily minimizers. Our methods
also apply to alternative models in the literature where the angular momentum
per unit mass is prescribed. The typical physical model our results apply to is
that of uniformly rotating white dwarf stars
The Volume of a Local Nodal Domain
Let M either be a closed real analytic Riemannian manifold or a closed smooth
Riemannian surface. We estimate from below the volume of a nodal domain
component in an arbitrary ball provided that this component enters the ball
deeply enough.Comment: 21 pages; introduction improved putting the problem in a larger
context
\epsilon-regularity for systems involving non-local, antisymmetric operators
We prove an epsilon-regularity theorem for critical and super-critical
systems with a non-local antisymmetric operator on the right-hand side.
These systems contain as special cases, Euler-Lagrange equations of
conformally invariant variational functionals as Rivi\`ere treated them, and
also Euler-Lagrange equations of fractional harmonic maps introduced by Da
Lio-Rivi\`ere.
In particular, the arguments presented here give new and uniform proofs of
the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and
also the integrability results by Sharp-Topping and Sharp, not discriminating
between the classical local, and the non-local situations
Nonlinear Dynamical Stability of Newtonian Rotating White Dwarfs and Supermassive Stars
We prove general nonlinear stability and existence theorems for rotating star
solutions which are axi-symmetric steady-state solutions of the compressible
isentropic Euler-Poisson equations in 3 spatial dimensions. We apply our
results to rotating and non-rotating white dwarf, and rotating high density
supermassive (extreme relativistic) stars, stars which are in convective
equilibrium and have uniform chemical composition. This paper is a continuation
of our earlier work ([28])
First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
We calculate the first and the second variation formula for the
sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We
consider general variations that can move the singular set of a C^2 surface and
non-singular variation for C_H^2 surfaces. These formulas enable us to
construct a stability operator for non-singular C^2 surfaces and another one
for C2 (eventually singular) surfaces. Then we can obtain a necessary condition
for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in
term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally
we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly
changed and Remark 9.9 adde
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