Attainability of the fractional Hardy constant with nonlocal mixed boundary conditions. Applications

The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \textit{fractional Hardy inequality } $$\Lambda_{N}\equiv\Lambda_{N}(\Omega):=\inf_{\{\phi\in \mathbb{E}^s(\Omega, D), \phi\neq 0\}} \dfrac{\frac{a_{d,s}}{2} \displaystyle\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \dfrac{|\phi(x)-\phi(y)|^2}{|x-y|^{d+2s}}dx dy} {\displaystyle\int_\Omega \frac{\phi^2}{|x|^{2s}}\,dx},$$ where $\Omega$ is a bounded domain of $\mathbb{R}^d$, $0<s<1$, $D\subset \mathbb{R}^d\setminus \Omega$ a nonempty open set and $$\mathbb{E}^{s}(\Omega,D)=\left\{ u \in H^s(\mathbb{R}^d):\, u=0 \text{ in } D\right\}.$$ The second aim of the paper is to study the \textit{mixed Dirichlet-Neumann boundary problem} associated to the minimization problem and related properties; precisely, to study semilinear elliptic problem for the \textit{fractional laplacian}, that is, $$P_{\lambda} \, \equiv \left\{ \begin{array}{rcll} (-\Delta)^s u &= & \lambda \dfrac{u}{|x|^{2s}} +u^p & {\text{ in }}\Omega, u & > & 0 &{\text{ in }} \Omega, \mathcal{B}_{s}u&:=&u\chi_{D}+\mathcal{N}_{s}u\chi_{N}=0 &{\text{ in }}\mathbb{R}^{d}\backslash \Omega, \\ \end{array}\right.$$ with $N$ and $D$ open sets in $\mathbb{R}^d\backslash\Omega$ such that $N \cap D=\emptyset$ and $\overline{N}\cup \overline{D}= \mathbb{R}^d \backslash\Omega$, $d>2s$, $\lambda> 0$ and $0<p\le 2_s^*-1$, $2_s^*=\frac{2d}{d-2s}$. We emphasize that the nonlinear term can be critical. The operators $(-\Delta)^s$, fractional laplacian, and $\mathcal{N}_{s}$, nonlocal Neumann condition, are defined below in (1.5) and (1.6) respectively

Catalytic Pyrolysis of Olive Mill Wastewater Sludge

Olive mill wastewater sludge (OMWS) is the solid residue that remains in the evaporation ponds after evaporation of the majority of water in the olive mill wastewater (OMW). OMWS is a major environmental pollutant in the olive oil producing regions. Approximately 41.16 wt. % of the OMWS was soluble in hexanes (HSF). The fatty acids in this fraction consist mainly of oleic and palmitic acid. Catalytic pyrolysis of the OMWS over red mud and HZSM-5 has been demonstrated to be an effective technology for converting this waste material into fuel. Red mud-catalyzed pyrolysis gave higher organics yields than the HZSM-5 catalysis. The viscosity as well as the oxygen content of the catalytic pyrolysis oils were significantly lower than those of the non-catalytic oil. The reaction pathways of red mud and HZSM-5 were different. The catalytic pyrolysis of the HSF gave an acidic oil with low viscosity and high energy content, and was nitrogen and sulfur free, whereas the catalytic pyrolysis of the solid residue after hexanes extraction (SR) gave an oil with higher viscosity, close to neutral pH, lower energy content, and had high nitrogen content and traces of sulfur