Existence Results For Semilinear Problems in the Two Dimensional Hyperbolic Space Involving Critical Growth

We consider semilinear elliptic problems on two-dimensional hyperbolic space involving critical growth. We first establish the Palais-Smale(P-S) condition and using (P-S) condition we obtain existence of solutions. In addition, we also explore existence of infinitely many sign changing solutions as well.Comment: Some new results adde

OVERVIEW AND PROSPECTS FOR DEVELOPMENT OF WAVE AND OFFSHORE WIND ENERGY

An overview of the present state of development of offshore renewable wave and wind energy is presented and future prospects are discussed. The information on some of the current wave energy systems worldwide are given as indicative of the present state of affairs. The main working principles of wave energy systems are described and the differences in terms of working principle, conversion chain, location and power take-off systems are highlighted. Some of the technology challenges are identified and the prospects of utilization of the various wave energy concepts are discussed comparing the characteristics of the devices in particular their power output. The evolution of the concepts of wind turbines with time and the main types of offshore wind turbine concepts are presented, from the shallow water fixed ones to the floating ones. The development of various numerical codes for the dynamic analysis of offshore wind turbines and the studies carried out based on the codes for hydrodynamic, aerodynamic, structural and response due to control system are presented. The present status of wind energy compared to wave energy and the role of naval architects and ocean engineers for the design and analysis of wave energy device and offshore wind turbine technology are presented and discussed

Sharp quantitative stability of Struwe's decomposition of the Poincar\'e-Sobolev inequalities on the hyperbolic space: Part I

A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space $$-\Delta_{\mathbb{B}^n} u-\lambda u = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n),$$ are unique up to hyperbolic isometries where $n \geq 3,$ $1 < p \leq \frac{n+2}{n-2}$ and $\lambda \leq \frac{(n-1)^2}{4}.$ We prove under certain bounds on $\|\nabla u \|_{L^2(\mathbb{B}^n)}$ the inequality $$\delta(u) \lesssim \|\Delta_{\mathbb{B}^n} u+ \lambda u + u^{p}\|_{H^{-1}},$$ holds whenever $p >2$ and hence forcing the dimensional restriction $3 \leq n \leq 5,$ where $\delta(u)$ denotes the $H^1$ distance of $u$ from the manifold of sums of hyperbolic bubbles. Moreover, it fails for any $n \geq 3$ and $p \in (1,2].$ This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent $p$ is $>2$ or $\leq 2$. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration. Mech. Anal, 237 (2020)] but we notice a striking dependence on the exponent $p$ in the subcritical regime as well which is not present in the flat case. Our technique is an amalgamation of Figalli and Glaudo's method and builds upon a series of new and novel estimates on the interaction of hyperbolic bubbles and their derivatives and improved eigenfunction integrability estimates. Since the conformal group coincides with the isometry group of the hyperbolic space, we perceive a remarkable distinction in arguments and techniques to achieve our main results compared to that of the Euclidean case.Comment: 70 pages, 4 figures. This is the updated version of our previous submission arXiv:2211.14618. New results have been added e.g., Section 9 and Section 10. The main new result is contained in Theorem~9.

Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p, \lambda}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{{p+1}} \, {\rm d}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-\lambda u^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\in C_c^{\infty}(\mathbb{B}^n),$ and $\lambda \leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb{B}^n}.$ It is known from the results of Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriate assumptions on $n,p$ and $\lambda$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,\lambda}(\mathbb{B}^n).$ In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in $\mathbb{R}^n$ of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincar\'{e}-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity

Thermodynamic and kinetic studies on novel Platinum(II) Complex containing bidentate N,N-donor ligands in ethanol-water medium

760-767Kinetic and mechanistic investigations have been made on the displacement of the two aqua molecules from the complex 1 i.e., [Pt(2,2′-bipyridine)(H2O)2]2+ represented as {Pt(bpy)} (bpy = 2,2′-bipyridine) at pH 4.0. All the substitution reactions have been monitored at 264, 284 & 317 nm, where the spectral difference between the reactant and product is maximum. Two consecutive reaction steps have been observed for the substitution of aqua molecules with some bidentate N,N-donor ligands namely, dimethylglyoxime (L1H), 1, 2-cyclohexanedionedioxime (L2H) and α-furildioxime (L3H) in ethanol-water medium using variable-temperature and stopped-flow spectrophotometry. Among the two steps, the former is ligand dependent and the later is ligand independent, where chelation is observed. All rate and activation parameters are consistent with associative substitution mechanisms. The thermodynamic parameters have been also calculated, which gives a negative DG0 value at all temperatures studied, supporting the spontaneous formation of an outer sphere association complex. The products of the reaction have been characterized with the help of IR and ESI-MS spectroscopic analysis

Measurement of t(t)over-bar normalised multi-differential cross sections in pp collisions at root s=13 TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions

Peer reviewe