The mixed problem in Lipschitz domains with general decompositions of the boundary

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let $\Lambda$ denote the boundary of $D$ (relative to $\partial\Omega$) and impose conditions on the dimension and shape of $\Lambda$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $\Omega$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D)$ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\partial\Omega)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.Comment: 36 page

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo

Radio continuum emission in the northern Galactic plane: Sources and spectral indices from the THOR survey

Context. Radio continuum surveys of the Galactic plane can find and characterize Hii regions, supernova remnants (SNRs), planetary nebulae (PNe), and extragalactic sources. A number of surveys at high angular resolution (≤2500) at different wavelengths exist to study the interstellar medium (ISM), but no comparable high-resolution and high-sensitivity survey exists at long radio wavelengths around 21 cm. Aims. Our goal is to investigate the 21 cm radio continuum emission in the northern Galactic plane at <2500 resolution. Methods.We observed a large percentage of the Galactic plane in the first quadrant of the MilkyWay (l = 14:0..67:4° and jbj ≤ 1:25°) with the Karl G. Jansky Very Large Array (VLA) in the C-configuration covering six continuum spectral windows (SPW). These data provide a detailed view on the compact as well as extended radio emission of our Galaxy and thousands of extragalactic background sources. Results.We used the BLOBCAT software and extracted 10 916 sources. After removing spurious source detections caused by the side lobes of the synthesized beam, we classified 10 387 sources as reliable detections. We smoothed the images to a common resolution of 2500 and extracted the peak flux density of each source in each SPW to determine the spectral indices (assuming I(v)/∞). By cross-matching with catalogs of Hii regions, SNRs, PNe, and pulsars, we found radio counterparts for 840 Hii regions, 52 SNRs, 164 PNe, and 38 pulsars. We found 79 continuum sources that are associated with X-ray sources. We identified 699 ultra-steep spectral sources (1:3) that could be high-redshift galaxies. About 9000 of the sources we extracted are not classified specifically, but based on their spatial and spectral distribution, a large percentage of these are likely to be extragalactic background sources. More than 7750 sources do not have counterparts in the SIMBAD database and more than 3760 sources do not have counterparts in the NED database. Conclusions. Studying the long wavelengths centimeter continuum emission and the associated spectral indices allows us to characaterize a large percentage of Galactic and extragalactic radio sources in the area of the northern inner Milky Way. This database will be extremely useful for future studies of a diverse set of astrophysical objects.The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Y. W., H. B., S. B., and J. D. S. acknowledge support from the European Research Council under the Horizon 2020 Framework Program via the ERC Consolidator Grant CSF-648505, and RSK via the ERC AdvancedGrant 339177 (STARLIGHT). H.B. and M.R. acknowledge support from the Deutsche Forschungsgemeinschaft in the Collaborative Research Center (SFB 881) “The Milky Way 74 System” (subproject B1, B2, B8), and from the Priority Program SPP 1573 “Physics of the Interstellar Medium” (grant numbers KL 1358/18.1, KL 1358/19.2). F.B. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme (grant agreement No 726384). The authors thank J. Mottram, M. Fouesneau and K. Meisenheimer for productive and fruitful discussions. This research made use of Astropy and affiliated packages, a community-developed core Python package for Astronomy (Astropy Collaboration 2018), Python package SciPy, APLpy, an open-source plotting package for Python (Robitaille & Bressert 2012), and software TOPCAT (Taylor 2005)

CMZoom III: Spectral Line Data Release

We present an overview and data release of the spectral line component of the SMA Large Program, \textit{CMZoom}. \textit{CMZoom} observed $^{12}$CO(2-1), $^{13}$CO(2-1) and C$^{18}$O(2-1), three transitions of H$_{2}$CO, several transitions of CH$_{3}$OH, two transitions of OCS and single transitions of SiO and SO, within gas above a column density of N(H$_2$)$\ge 10^{23}$\,cm$^{-2}$ in the Central Molecular Zone (CMZ; inner few hundred pc of the Galaxy). We extract spectra from all compact 1.3\,mm \emph{CMZoom} continuum sources and fit line profiles to the spectra. We use the fit results from the H$_{2}$CO 3(0,3)-2(0,2) transition to determine the source kinematic properties. We find $\sim 90$\% of the total mass of \emph{CMZoom} sources have reliable kinematics. Only four compact continuum sources are formally self-gravitating. The remainder are consistent with being in hydrostatic equilibrium assuming that they are confined by the high external pressure in the CMZ. Based on the mass and density of virially bound sources, and assuming star formation occurs within one free-fall time with a star formation efficiency of $10\% - 75\%$, we place a lower limit on the future embedded star-formation rate of $0.008 - 0.06$\,M$_{\odot}$\,yr$^{-1}$. We find only two convincing proto-stellar outflows, ruling out a previously undetected population of very massive, actively accreting YSOs with strong outflows. Finally, despite having sufficient sensitivity and resolution to detect high-velocity compact clouds (HVCCs), which have been claimed as evidence for intermediate mass black holes interacting with molecular gas clouds, we find no such objects across the large survey area.Comment: 44 pages, 41 figure

Personalizing health care: feasibility and future implications

Considerable variety in how patients respond to treatments, driven by differences in their geno- and/ or phenotypes, calls for a more tailored approach. This is already happening, and will accelerate with developments in personalized medicine. However, its promise has not always translated into improvements in patient care due to the complexities involved. There are also concerns that advice for tests has been reversed, current tests can be costly, there is fragmentation of funding of care, and companies may seek high prices for new targeted drugs. There is a need to integrate current knowledge from a payer’s perspective to provide future guidance. Multiple findings including general considerations; influence of pharmacogenomics on response and toxicity of drug therapies; value of biomarker tests; limitations and costs of tests; and potentially high acquisition costs of new targeted therapies help to give guidance on potential ways forward for all stakeholder groups. Overall, personalized medicine has the potential to revolutionize care. However, current challenges and concerns need to be addressed to enhance its uptake and funding to benefit patients