Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space R+n\mathbb R_+^n

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$ $$\int_{\mathbb R_+^n} \int_{\partial \mathbb R_+^n} f(x) |x-y|^\lambda g(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\partial \mathbb R_+^n)} \, \|g\|_{L^r(\mathbb R_+^n)}$$ for any nonnegative functions $f\in L^p(\partial \mathbb R_+^n)$, $g\in L^r(\mathbb R_+^n)$, and $p,r\in (0,1)$, $\lambda > 0$ such that $(1-1/n)1/p + 1/r -(\lambda-1) /n =2$. Some estimates for $\mathscr C_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.Comment: 31 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:1508.02041. To appear in International Mathematics Research Notices (IMRN

A review on microalgae and cyanobacteria in biofuel production

Today, fossil fuel shortages and climate change impacts have led mankind to the search for an alternative energy. With many advantages, bioenergy is a promising source to replace conventional energy. However, biofuel productions in the first and second generation are likely to add more concerns to the problems in water scarcity and threats to food security. Meanwhile, third generation biofuels obtained from microalgae and cyanobacteria are able to overcome existing challenges thanks to their rapid growth rates, abilities to fix CO 2 , high yields in lipid extraction and capabilities to be grown in non-arable lands. Microalgae and cyanobacteria appear to be the only ones among renewable sources that are capable of producing a wide range of biofuels including biohydrogen, biomethane, bioethanol and biodiesel.In this study, we present an overview about microalgae and cyanobacteria use for the production of biofuels in fundamentals, including their biology, cultivation systems taking into account the hydrodynamic conditions, harvesting, and processing. The review also provides a general picture at the current status of this renewable energy industry

The Prospective Evolution of the Vietnamese Power Sector: The Vulnerability and Externality Analysis

With its rapidly increasing power demand of 16% p.a since 1990s along with its limited supply power capacity, how sustainable is Vietnam's electricity development? What are the major factors explaining its performance relative to other Asian countries? To answer these questions, this paper examines the Vietnamese power system from the 1990s to 2040 by using LEAP simulation. Twelve vulnerability and externality indexes regarding social-economic-environmental dimensions are calculated to assess vulnerability levels of the sector in seven scenarios. External costs of CO2, NOx, SO2 and PM10 are calculated to examine how far the costs could affect on the electricity cost in Vietnam. In sensitivity analysis, the paper assesses the impacts of international coal price's fluctuations on the electricity price and the trade balance of Vietnam. The study confirms that Vietnam's power sector will become more vulnerable to fossil fuels' prices, environmental pollutants and climate change if the sector goes for the current policy pathways. With the efficiency scenarios examined in the study, the sector would be more independent and less vulnerable. To reduce the vulnerabilities, the study suggests that Vietnam should promote energy efficiency and electricity generation from non-fossil fuels and internalize external costs into the power sector. Vietnam needs also get involved in international financial mechanisms for clean development and technology transfer programs to efficiently exploit its renewable energy potentials