1,730 research outputs found
Examples of compact Einstein four-manifolds with negative curvature
We give new examples of compact, negatively curved Einstein manifolds of
dimension . These are seemingly the first such examples which are not
locally homogeneous. Our metrics are carried by a sequence of 4-manifolds
previously considered by Gromov and Thurston. The construction begins
with a certain sequence of hyperbolic 4-manifolds, each containing a
totally geodesic surface which is nullhomologous and whose normal
injectivity radius tends to infinity with . For a fixed choice of natural
number , we consider the -fold cover branched along
. We prove that for any choice of and all large enough
(depending on ), carries an Einstein metric of negative sectional
curvature. The first step in the proof is to find an approximate Einstein
metric on , which is done by interpolating between a model Einstein metric
near the branch locus and the pull-back of the hyperbolic metric from .
The second step in the proof is to perturb this to a genuine solution to
Einstein's equations, by a parameter dependent version of the inverse function
theorem. The analysis relies on a delicate bootstrap procedure based on
coercivity estimates.Comment: 53 pages. v2 small modifications to exposition. v3 typos corrected.
Same text as published version, to appear in the Journal of the American
Mathematical Societ
Stability of Transonic Shocks in Steady Supersonic Flow past Multidimensional Wedges
We are concerned with the stability of multidimensional (M-D) transonic
shocks in steady supersonic flow past multidimensional wedges. One of our
motivations is that the global stability issue for the M-D case is much more
sensitive than that for the 2-D case, which requires more careful rigorous
mathematical analysis. In this paper, we develop a nonlinear approach and
employ it to establish the stability of weak shock solutions containing a
transonic shock-front for potential flow with respect to the M-D perturbation
of the wedge boundary in appropriate function spaces. To achieve this, we first
formulate the stability problem as a free boundary problem for nonlinear
elliptic equations. Then we introduce the partial hodograph transformation to
reduce the free boundary problem into a fixed boundary value problem near a
background solution with fully nonlinear boundary conditions for second-order
nonlinear elliptic equations in an unbounded domain. To solve this reduced
problem, we linearize the nonlinear problem on the background shock solution
and then, after solving this linearized elliptic problem, develop a nonlinear
iteration scheme that is proved to be contractive.Comment: 41 pages, 10 figure
Weak observability estimates for 1-D wave equations with rough coefficients
In this paper we prove observability estimates for 1-dimensional wave
equations with non-Lipschitz coefficients. For coefficients in the Zygmund
class we prove a "classical" observability estimate, which extends the
well-known observability results in the energy space for regularity. When
the coefficients are instead log-Lipschitz or log-Zygmund, we prove
observability estimates "with loss of derivatives": in order to estimate the
total energy of the solutions, we need measurements on some higher order
Sobolev norms at the boundary. This last result represents the intermediate
step between the Lipschitz (or Zygmund) case, when observability estimates hold
in the energy space, and the H\"older one, when they fail at any finite order
(as proved in \cite{Castro-Z}) due to an infinite loss of derivatives. We also
establish a sharp relation between the modulus of continuity of the
coefficients and the loss of derivatives in the observability estimates. In
particular, we will show that under any condition which is weaker than the
log-Lipschitz one (not only H\"older, for instance), observability estimates
fail in general, while in the intermediate instance between the Lipschitz and
the log-Lipschitz ones they can hold only admitting a loss of a finite number
of derivatives. This classification has an exact counterpart when considering
also the second variation of the coefficients.Comment: submitte
Skorohod and Stratonovich integration in the plane
This article gives an account on various aspects of stochastic calculus in
the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of
variable formula for a path indexed by a square, satisfying some H\"older
regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some
Skorohod change of variable formulas for a large class of Gaussian processes
defined on the suqare. (iii) Compare the bidimensional integrals obtained with
those two methods, computing explicit correction terms whenever possible. As a
byproduct, we also give explicit forms of corrections in the respective change
of variable formulas
Spectral Invariance of Non-Smooth Pseudodifferential Operators
In this paper we discuss some spectral invariance results for non-smooth
pseudodifferential operators with coefficients in H\"older spaces. In analogy
to the proof in the smooth case of Beals and Ueberberg, we use the
characterization of non-smooth pseudodifferential operators to get such a
result. The main new difficulties are the limited mapping properties of
pseudodifferential operators with non-smooth symbols and the fact, that in
general the composition of two non-smooth pseudodifferential operators is not a
pseudodifferential operator.
In order to improve these spectral invariance results for certain subsets of
non-smooth pseudodifferential operators with coefficients in H\"older spaces,
we improve the characterization of non-smooth pseudodifferential operators in a
previous work by the authors.Comment: 43 page
Conservation of geometric structures for non-homogeneous inviscid incompressible fluids
We obtain a result about propagation of geometric properties for solutions of
the non-homogeneous incompressible Euler system in any dimension . In
particular, we investigate conservation of striated and conormal regularity,
which is a natural way of generalizing the 2-D structure of vortex patches. The
results we get are only local in time, even in the dimension N=2; however, we
provide an explicit lower bound for the lifespan of the solution. In the case
of physical dimension N=2 or 3, we investigate also propagation of H\"older
regularity in the interior of a bounded domain
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