On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.Comment: 44 pages, introduction revised, references expanded. To appear in Arch. Rational Mech. Ana

Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.Comment: 24 pages, to be submitte

Partial Schauder estimates for second-order elliptic and parabolic equations

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H\"older semi-norms not with respect to all, but only with respect to some of the independent variables.Comment: CVPDE, accepted (2010)

UV-Optical Pixel Maps of Face-On Spiral Galaxies -- Clues for Dynamics and Star Formation Histories

UV and optical images of the face-on spiral galaxies NGC 6753 and NGC 6782 reveal regions of strong on-going star formation that are associated with structures traced by the old stellar populations. We make NUV--(NUV-I) pixel color-magnitude diagrams (pCMDs) that reveal plumes of pixels with strongly varying NUV surface brightness and nearly constant I surface brightness. The plumes correspond to sharply bounded radial ranges, with (NUV-I) at a given NUV surface brightness being bluer at larger radii. The plumes are parallel to the reddening vector and simple model mixtures of young and old populations, thus neither reddening nor the fraction of the young population can produce the observed separation between the plumes. The images, radial surface-brightness, and color plots indicate that the separate plumes are caused by sharp declines in the surface densities of the old populations at radii corresponding to disk resonances. The maximum surface brightness of the NUV light remains nearly constant with radius, while the maximum I surface brightness declines sharply with radius. An MUV image of NGC 6782 shows emission from the nuclear ring. The distribution of points in an (MUV-NUV) vs. (NUV-I) pixel color-color diagram is broadly consistent with the simple mixture model, but shows a residual trend that the bluest pixels in (MUV-NUV) are the reddest pixels in (NUV-I). This may be due to a combination of red continuum from late-type supergiants and [SIII] emission lines associated with HII regions in active star-forming regions. We have shown that pixel mapping is a powerful tool for studying the distribution and strength of on-going star formation in galaxies. Deep, multi-color imaging can extend this to studies of extinction, and the ages and metallicities of composite stellar populations in nearby galaxies.Comment: LaTeX with AASTeX style file, 29 pages with 12 figures (some color, some multi-part). Accepted for publication in The Astrophysical Journa

Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

We consider local "complementary" generalized Morrey spaces M-c({x0})p(.).omega (Omega) in which the p-means of function are controlled over Omega \ B(x(0), r) instead of B(x(0), r), where Omega subset of R-n is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function omega(r) defining the "complementary" Morrey-type norm. In the case where omega is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type M-c({x0})p(.).omega (Omega) -> M-c({x0})p(.).omega (Omega)-theorem for the potential operators I-alpha(.), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities-on omega(r), which do not assume any assumption on monotonicity of omega(r).Science Development Foundation under the President of the Republic of Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK) [110T695]info:eu-repo/semantics/publishedVersio

A novel role for the root cap in phosphate uptake and homeostasis

The root cap has a fundamental role in sensing environmental cues as well as regulating root growth via altered meristem activity. Despite this well-established role in the control of developmental processes in roots, the root cap's function in nutrition remains obscure. Here, we uncover its role in phosphate nutrition by targeted cellular inactivation or phosphate transport complementation in Arabidopsis, using a transactivation strategy with an innovative high-resolution real-time P-33 imaging technique. Remarkably, the diminutive size of the root cap cells at the root-to-soil exchange surface accounts for a significant amount of the total seedling phosphate uptake (approximately 20%). This level of Pi absorption is sufficient for shoot biomass production (up to a 180% gain in soil), as well as repression of Pi starvation-induced genes. These results extend our understanding of this important tissue from its previously described roles in environmental perception to novel functions in mineral nutrition and homeostasis control

Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a byproduct of our technique we are able to answer positively to a conjecture of E.B.Davies.Comment: 40 page

A limit model for thermoelectric equations

We analyze the asymptotic behavior corresponding to the arbitrary high conductivity of the heat in the thermoelectric devices. This work deals with a steady-state multidimensional thermistor problem, considering the Joule effect and both spatial and temperature dependent transport coefficients under some real boundary conditions in accordance with the Seebeck-Peltier-Thomson cross-effects. Our first purpose is that the existence of a weak solution holds true under minimal assumptions on the data, as in particular nonsmooth domains. Two existence results are studied under different assumptions on the electrical conductivity. Their proofs are based on a fixed point argument, compactness methods, and existence and regularity theory for elliptic scalar equations. The second purpose is to show the existence of a limit model illustrating the asymptotic situation.Comment: 20 page