Characterisation of upper gradients on the weighted Euclidean space and applications

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

Solar Pond Performance Enhances Nonconventional Water Resource Availability

In the context of utilizing solar ponds, this research was commenced to enhance their performance so as to rely on them as a nonconventional water resources. Primarily, literature was reviewed in the field of solar ponds. The technique can be used to develop the energy needed for pumping, lifting, collecting or treating water. The agricultural sector users claim that water and energy face problems of environmental degradation due to resources scarcity. This research paper, data was gathered and analyzed, in terms of solar pond parameters such as depth of the upper gradient, shading effect, storage zones, daylight hours, ground temperature and covered insulation for different climate zones so as for different latitudes. The analyzed results indicated that solar ponds possess high potential in arid and semi-arid climates similar to Fayoum governorate, where it is distinguished by its ability to collect heat which can be utilized in different applications. The application cover desalination, electric power generation, salt purification, food and fishing industries. The solar pond technique utilized in many countries to act as the backbone for sustainable development in arid and semi-arid zones such as Victoria desert in Australia.

Newtonian Spaces Based on Quasi-Banach Function Lattices

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.