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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]

Let Ω ⊂ ℝN (N ≥ 2) be a bounded domain with smooth boundary. We show the existence of a positive real number λ* such that for each λ ∈ (0, λ*) and each real number p > N the equation −Δp u = λeu in Ω subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution up. Next, we analyze the asymptotic behavior of up as p → ∞ and we show that it converges uniformly to the distance function to the boundary of the domain. [ABSTRACT FROM AUTHOR]

We study the boundary value problem --div((|∇u|p1(x)-2+|∇u |p2(x)-2)∇u)= f(x,u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝN. Our attention is focused on two cases when f(x, u) = ±(-λ|u|m(x)-2u + |u|q(x)-2u), where m(x) = max{p1(x), p2(x)} for any x ∈ Ω or m(x) < q(x) < N ⋅ m(x)/(N - m(x)) for any x ∈ Ω. In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. [ABSTRACT FROM AUTHOR]