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Let $D>1$ be a fixed integer. Given a smooth bounded, convex domain $\Omega \subset \mathbb{R}^D$ and $H:\mathbb{R}^D\rightarrow [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }(\mathbb{R}^D\setminus \lbrace 0\rbrace )$ for which the Hessian matrix $D^2(H^p)$ is positive definite in $\mathbb{R}^D\setminus \lbrace 0\rbrace $ for any $p\in (1,\infty )$, we study the monotonicity of the principal frequency of the anisotropic $p$-Laplacian (constructed using the function $H$) on $\Omega $ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega $ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $\mathbb{R}^D$, we recover some recent results obtained by the first two authors in [3, 4].

One of the very important parameters for the optimal growth and development of vegetation is precipitation. In the Moldova River upper basin no recent studies have been conducted on the climatic elements that are important for living organisms and human activities. The most representative were made by Barbu N. (1976), Erhan E. (1989), Apostol L. and Rusu C. (1988), Rusu C. 2002 and the researches of Barbu I. and Popa I., ICAS Câmpulung (1990, 2003). The present paper makes an analysis of precipitation measured at the two stations in the area: Rarău, where rainfall is still currently measured, and Câmpulung Moldovenesc, where measurements had been conducted until 2000.

Energy management is important for all farms and choosing the most suitable form of storing energy is essential in mitigating associated risks, seizing opportunities and thus maximizing productivity. We propose the small scale environmentally friendly method of storing energy in gravitational form, which involves pumping water available in open ponds to higher altitudes to store surplus energy, and releasing it to lower altitudes when additional energy is required. For small farms that benefit of suitable geographical position for storing energy this way, i.e. sufficiently high height gradient, this method would be particularly beneficial as it decreases the need to rely on external energy providers. In addition, it has the potential of helping the environment, as local flora and fauna may benefit from the presence of water especially during drought periods. For the purpose of remote control and data acquisition in such locations, we propose using long range and low power communication (LoRa) infrastructure and LoraWAN protocol - such infrastructure has the added benefit of assisting farmers in making better decisions for their crops, as well as providing warnings if required. To assess the feasibility of the method, we carried out a case study on a local small farm. Initial results indicate that the costs involved when using this method are comparable to other energy storage methods, but storing energy gravitationally has the added advantage of being more environmentally friendly. For a more detailed and larger scale feasibility test, we propose to use Artificial Intelligence (AI) for choosing the most suitable locations for placing the ponds and Machine Learning (ML) techniques to examine correlations and draw a conclusion on whether the presence of new ponds is overall beneficial. Such an investigation also offers insight from a quantitative perspective, informing economic calculations such as pay-back period (PBP) and Return on Investment (ROI) of the project. [ABSTRACT FROM AUTHOR]

We investigate some monotonicity properties of the best constant from Morrey's inequality in convex and bounded domains from the Euclidean space \mathbb R^D (D\geq 1). Using these monotonicity properties we give a new variational characterization of the best constant from Morrey's inequality on convex and bounded domains for which the maximum of the distance function to the boundary is small. We also show that this variational characterization does not hold true on convex and bounded domains for which the maximum of the distance function to the boundary is larger than 1. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of the sequence of solutions for some families of torsional creep-type problems, subject to the homogenous Dirichlet boundary conditions has been a very active field of investigation throughout the years. In this survey we present several results in the field. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of Rayleigh quotients involving Luxemburg norms in Orlicz spaces L Ψ n is studied as n → ∞ , when (Ψ n) n is a sequence of N-functions. In a particular case, we recover a well-known result of Juutinen, Lindqvist & Manfredi and Fukagai, Ito & Narukawa regarding the limit, as p → ∞ , of the minima of Rayleigh quotients associated to the eigenvalue problem for the p-Laplacian. [ABSTRACT FROM AUTHOR]

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω. [ABSTRACT FROM AUTHOR]

The study of perturbed eigenvalue problems has been a very active field of investigation throughout the years. In this survey we collect several results in the field. [ABSTRACT FROM AUTHOR]

For any fixed integer D > 1 we show that there exists M ∈ [e−1, 1 ] such that for any open, bounded, convex domain Ω ⊂ ℝD with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on (1 , ∞). Moreover, for any real number s > M there exists an open, bounded, convex domain Ω ⊂ ℝD with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of p ∈ (1 , ∞). [ABSTRACT FROM AUTHOR]

Let D > 1 be a positive integer. For each real numbers a and b such that 0 < a < b < ∞ consider the annulus A(a, b) := {x ∈ RD : a < |x| < b}. Our goal is to give sufficient conditions on a and b such that the function which gives the principal eigenvalue of the p -Laplacian on A(a, b), subject to the homogeneous Dirichlet boundary condition, be monotone or non-monotone with respect to p ∈ (1, ∞). [ABSTRACT FROM AUTHOR]

We deal with monotonicity with respect to pp of the first positive eigenvalue of the pp-Laplace operator on Ω subject to the homogeneous Neumann boundary condition. For any fixed integer D > 1 we show that there exists M∈[2e−1,2] such that for any open, bounded, convex domain Ω⊂RD with smooth boundary for which the diameter of Ω is less than or equal to M, the first positive eigenvalue of the p-Laplace operator on Ω subject to the homogeneous Neumann boundary condition is an increasing function of p on (1,∞). Moreover, for each real number s > M there exists a sequence of open, bounded, convex domains {Ωn}n⊂RD with smooth boundaries for which the sequence of the diameters of Ωn converges to s, as n→∞, and for each n large enough the first positive eigenvalue of the p-Laplace operator on Ωn subject to the homogeneous Neumann boundary condition is not a monotone function of p on (1,∞). [ABSTRACT FROM AUTHOR]