480 research outputs found
Weak UCP and perturbed monopole equations
We give a simple proof of weak Unique Continuation Property for perturbed
Dirac operators, using the Carleman inequality. We apply the result to a class
of perturbations of the Seiberg-Witten monopole equations that arise in Floer
theory.Comment: 22 pages LaTeX, one .eps figur
Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform
We extend the -theory of the Boltzmann collision operator by using
classical techniques based in the Carleman representation and Fourier analysis,
allied to new ideas that exploit the radial symmetry of this operator. We are
then able to greatly simplify existent technical proofs in this theory, extend
the range, and obtain explicit sharp constants in some convolution-like
inequalities for the gain part of the Boltzmann collision operator.Comment: 14 page
On quasianalytic local rings
This expository article is devoted to the local theory of ultradifferentiable
classes of functions, with a special emphasis on the quasianalytic case.
Although quasianalytic classes are well-known in harmonic analysis since
several decades, their study from the viewpoint of differential analysis and
analytic geometry has begun much more recently and, to some extent, has earned
them a new interest. Therefore, we focus on contemporary questions closely
related to topics in local algebra. We study, in particular, Weierstrass
division problems and the role of hyperbolicity, together with properties of
ideals of quasianalytic germs. Incidentally, we also present a simplified proof
of Carleman's theorem on the non-surjectivity of the Borel map in the
quasianalytic case.Comment: Final Manuscrip
Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions
We consider a second-order selfadjoint elliptic operator with an anisotropic
diffusion matrix having a jump across a smooth hypersurface. We prove the
existence of a weight-function such that a Carleman estimate holds true. We
moreover prove that the conditions imposed on the weight function are
necessary.Comment: 43 page
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
Strong unique continuation for general elliptic equations in 2D
We prove that solutions to elliptic equations in two variables in divergence
form, possibly non-selfadjoint and with lower order terms, satisfy the strong
unique continuation property.Comment: 10 page
Explicit coercivity estimates for the linearized Boltzmann and Landau operators
We prove explicit coercivity estimates for the linearized Boltzmann and
Landau operators, for a general class of interactions including any
inverse-power law interactions, and hard spheres. The functional spaces of
these coercivity estimates depend on the collision kernel of these operators.
They cover the spectral gap estimates for the linearized Boltzmann operator
with Maxwell molecules, improve these estimates for hard potentials, and are
the first explicit coercivity estimates for soft potentials (including in
particular the case of Coulombian interactions). We also prove a regularity
property for the linearized Boltzmann operator with non locally integrable
collision kernels, and we deduce from it a new proof of the compactness of its
resolvent for hard potentials without angular cutoff.Comment: 32 page
Hardy-Carleman Type Inequalities for Dirac Operators
General Hardy-Carleman type inequalities for Dirac operators are proved. New
inequalities are derived involving particular traditionally used weight
functions. In particular, a version of the Agmon inequality and Treve type
inequalities are established. The case of a Dirac particle in a (potential)
magnetic field is also considered. The methods used are direct and based on
quadratic form techniques
On a Watson-like Uniqueness Theorem and Gevrey Expansions
We present a maximal class of analytic functions, elements of which are in
one-to-one correspondence with their asymptotic expansions. In recent decades
it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.),
that the formal power series solutions of a wide range of systems of ordinary
(even non-linear) analytic differential equations are in fact the Gevrey
expansions for the regular solutions. Watson's uniqueness theorem belongs to
the foundations of this new theory. This paper contains a discussion of an
extension of Watson's uniqueness theorem for classes of functions which admit a
Gevrey expansion in angular regions of the complex plane with opening less than
or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We
present conditions which ensure uniqueness and which suggest an extension of
Watson's representation theorem. These results may be applied for solutions of
certain classes of differential equations to obtain the best accuracy estimate
for the deviation of a solution from a finite sum of the corresponding Gevrey
expansion.Comment: 18 pages, 4 figure
A new look at the pair-production width in a strong magnetic field
We reexamine the process in a background magnetic field
comparable to . This process is known to be non-perturbative
in the magnetic-field strength. However, it can be shown that the {\it moments}
of the above pair production width is proportional to the derivatives of photon
polarization function at the zero energy, which is perturbative in . Hence,
the pair-production width can be easily obtained from the latter by the inverse
Mellin transform. The implications of our approach are discussed.Comment: 8 pages, 2 figures, Revtex, Some comments adde
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