428 research outputs found
The classical overdetermined Serrin problem
In this survey we consider the classical overdetermined problem which was
studied by Serrin in 1971. The original proof relies on Alexandrov's moving
plane method, maximum principles, and a refinement of Hopf's boundary point
Lemma. Since then other approaches to the same problem have been devised. Among
them we consider the one due to Weinberger which strikes for the elementary
arguments used and became very popular. Then we discuss also a duality approach
involving harmonic functions, a shape derivative approach and a purely integral
approach, all of them not relying on maximum principle. For each one we
consider pros and cons as well as some generalizations
On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets
Let be an open convex set in with finite width, and
let be the torsion function for , i.e. the solution of
. An upper bound is obtained for the product
of , where
is the bottom of the spectrum of the Dirichlet Laplacian
acting in . The upper bound is sharp in the limit of a thinning
sequence of convex sets. For planar rhombi and isosceles triangles with area
, it is shown that , and that this bound is sharp.Comment: 12 pages, 4 figure
Average fractional polarization of extragalactic sources at Planck frequencies
Recent detailed simulations have shown that an insufficiently accurate
characterization of the contamination of unresolved polarized extragalactic
sources can seriously bias measurements of the primordial cosmic microwave
background (CMB) power spectrum if the tensor-to-scalar ratio as
predicted by models currently of special interest (e.g., Starobinsky's
and Higgs inflation). This has motivated a reanalysis of the median
polarization fraction of extragalactic sources (radio-loud AGNs and dusty
galaxies) using data from the \textit{Planck} polarization maps. Our approach,
exploiting the intensity distribution analysis, mitigates or overcomes the most
delicate aspects of earlier analyses based on stacking techniques. By means of
simulations, we have shown that the residual noise bias on the median
polarization fraction, , of extragalactic sources is
generally \simlt 0.1\%. For radio sources, we have found , with no significant dependence on either frequency or flux
density, in good agreement with the earlier estimate and with high-sensitivity
measurements in the frequency range 5--40\,GHz. No polarization signal is
detected in the case of dusty galaxies, implying 90\% confidence upper limits
of \Pi_{\rm dusty}\simlt 2.2\% at 353\,GHz and of \simlt 3.9\% at 217\,GHz.
The contamination of CMB polarization maps by unresolved point sources is
discussed.Comment: 10 pages, 3 figures, 7 tables; revised version. In press on Astronomy
and Astrophysic
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