Boundary Harnack estimates in slit domains and applications to thin free boundary problems

We provide a higher order boundary Harnack inequality for harmonic functions in slit domains. As a corollary we obtain the $C^\infty$ regularity of the free boundary in the Signorini problem near non-degenerate points

The two membranes problem for different operators

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the H\"older continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain $C^{1,\gamma}$ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary

Is H3+ cooling ever important in primordial gas?

Studies of the formation of metal-free Population III stars usually focus primarily on the role played by H2 cooling, on account of its large chemical abundance relative to other possible molecular or ionic coolants. However, while H2 is generally the most important coolant at low gas densities, it is not an effective coolant at high gas densities, owing to the low critical density at which it reaches local thermodynamic equilibrium (LTE) and to the large opacities that develop in its emission lines. It is therefore possible that emission from other chemical species may play an important role in cooling high density primordial gas. A particularly interesting candidate is the H3+ molecular ion. This ion has an LTE cooling rate that is roughly a billion times larger than that of H2, and unlike other primordial molecular ions such as H2+ or HeH+, it is not easily removed from the gas by collisions with H or H2. It is already known to be an important coolant in at least one astrophysical context -- the upper atmospheres of gas giants -- but its role in the cooling of primordial gas has received little previous study. In this paper, we investigate the potential importance of H3+ cooling in primordial gas using a newly-developed H3+ cooling function and the most detailed model of primordial chemistry published to date. We show that although H3+ is, in most circumstances, the third most important coolant in dense primordial gas (after H2 and HD), it is nevertheless unimportant, as it contributes no more than a few percent of the total cooling. We also show that in gas irradiated by a sufficiently strong flux of cosmic rays or X-rays, H3+ can become the dominant coolant in the gas, although the size of the flux required renders this scenario unlikely to occur.Comment: 60 pages, 22 figures. Submitted to MNRA

Evaluating Rice Straw as a Substitute for Barley Straw in Inhibiting Algal Growth in Farm Ponds

Algal blooms disrupt aquatic ecosystems and are more common in lakes, ponds, and rivers during the summer months due to nutrient pollution. Livestock production can contribute increased quantities of nutrients to water bodies from runoff of manure. Commonly used mechanical and chemical control methods may have limited success because algae are small and propagate quickly. Barley (Hordeum vulgare) straw has been shown to inhibit the growth of algae as the straw decomposes aerobically in ponds. Therefore, barley represents a natural option for algal biomass control. However, the small amount of barley production in Arkansas limits the availability of barley straw as a solution to control algal blooms locally. Other cereal grain straws may produce similar inhibitory effects during decomposition. Rice (Oryza sativa) is produced in large quantities in Arkansas, making rice straw a locally sourced straw product. The objective of this research was to determine the efficacy of using rice compared to barley straw to inhibit algal growth in freshwater ponds. Data were collected from nine farm ponds, three treated with rice straw, three treated with barley straw, and three without amendment to serve as the experimental control. Dissolved oxygen, pH, nitrate-nitrogen (NO3--N), dissolved phosphorus (P), temperature, and turbidity were measured for 14 weeks from June 12 to September 17, 2018. Algal biomass was measured as chlorophyll-a concentration to evaluate treatment effectiveness over time. Dissolved oxygen was significantly influenced by treatment and time. The NO3--N concentration in ponds treated with rice straw was significantly greater than the control and barley treatment. Chlorophyll-a concentrations were variable, and there were no consistent trends through time within a treatment. More research under controlled conditions to understand impacts of abiotic conditions, microbial and algal community compositions, and mode of action of algal inhibition is required before cereal straw can be a reliable, locally sourced method of algal control in farm ponds

Normal heat conductivity in two-dimensional scalar lattices

The paper revisits recent counterintuitive results on divergence of heat conduction coefficient in two-dimensional lattices. It was reported that in certain lattices with on-site potential, for which one-dimensional chain has convergent conductivity, for the 2D case it turns out to diverge. We demonstrate that this conclusion is an artifact caused by insufficient size of the simulated system. To overcome computational restrictions, a ribbon of relatively small width is simulated instead of the square specimen. It is further demonstrated that the heat conduction coefficient in the "long" direction of the ribbon ceases to depend on the width, as the latter achieves only 10 to 20 interparticle distances. So, one can consider the dynamics of much longer systems, than in the traditional setting, and still can gain a reliable information regarding the 2D lattice. It turns out that for all considered models, for which the conductivity is convergent in the 1D case, it is indeed convergent in the 2D case. In the same time, however, the length of the system, necessary to reveal the convergence in the 2D case, may be much bigger than in its 1D counterpart.Comment: 6 pages, 6 figure

Heat Conduction in One-Dimensional chain of Hard Discs with Substrate Potential

Heat conduction of one-dimensional chain of equivalent rigid particles in the field of external on-site potential is considered. Zero diameters of the particles correspond to exactly integrable case with divergent heat conduction coefficient. By means of simple analytical model it is demonstrated that for any nonzero particle size the integrability is violated and the heat conduction coefficient converges. The result of the analytical computation is verified by means of numerical simulation in a plausible diapason of parameters and good agreement is observedComment: 14 pages, 7 figure