Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent

Given a smooth bounded domain [Formula: see text] in [Formula: see text] with [Formula: see text], we study the existence and the profile of positive solutions for the following elliptic Neumann problem: [Formula: see text] where [Formula: see text] is a large exponent and [Formula: see text] denotes the outer unit normal vector to the boundary [Formula: see text]. For suitable domains [Formula: see text], by a constructive way we prove that, for any non-negative integers [Formula: see text], [Formula: see text] with [Formula: see text], if [Formula: see text] is large enough, such a problem has a family of positive solutions with [Formula: see text] boundary layers and [Formula: see text] interior layers which concentrate along [Formula: see text] distinct [Formula: see text]-dimensional minimal submanifolds of [Formula: see text], or collapse to the same [Formula: see text]-dimensional minimal submanifold of [Formula: see text] as [Formula: see text].

Variational Approaches to P(X)-Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena

This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena, $$\begin{equation*} \left\{\begin{array}{ll} -{\rm div}\bigg(\bigg(1+\frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\bigg) |\nabla u|^{p(x)-2}\nabla u\bigg)+\alpha(x)|u|^{p(x)-2}u\\=\lambda f(x,u) \mbox{in}\,\,\Omega,\\ \frac{\partial u}{\partial \nu}=0\mbox{on}\,\,\partial \Omega \end{array}\right. \end{equation*}$$where $\Omega \subset \mathbb{R}^N$$(N\geq 2)$ is a bounded domain with boundary of class $C^1,$$\nu$ is the outer unit normal to $\partial \Omega,$$\lambda>0$, $\alpha\in L^{\infty}(\Omega),$$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function and $p\in C^0(\overline{\Omega})$. Our technical approach is based on variational methods and we use a more precise version of Ricceri’s Variational Principle due to Bonanno and Molica Bisci. Some recent results are extended and improved. Some examples are presented to illustrate the application of our main results.

Clustering layers for the Fife—Greenlee problem in ℝn

We consider the following Fife–Greenlee problem: where Ω is a smooth and bounded domain in ℝn, ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a(x) ∈ (–1, 1) in . Let K, Ω– and Ω+ be the zero-level sets of a, {a < 0} and {a < 0}, respectively. We assume ∇a ≠ 0 on K. Fife and Greenlee constructed stable layer solutions, while del Pino et al. proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = εj → 0, and a solution with m-transition layers near K. The distance of any two layers is O(ε log(1/ε)). Furthermore, converges uniformly to ±1 on the compact sets of Ω± as j → +∞

Extension of the unit normal vector field from a hypersurface

In many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

Cytoplasmic segregation in onion root tip cells and formation of vacuoles

Onion root tips were studied both with transmission and scanning electron microscope. The cells in the meristematic region of initials were found to be little vacuolated. At early stages of cell differentiation areas free of ribosomes are segregated in the cytoplasm. No (or only incomplete) membranes delimiting such areas could be distinguished in most cases. Seemingly, the membraneous structures may form later or simultaneously with the progress of breakdown of biostructure of the cytoplasm in the segregated area. The microscopic image of cytoplasm in the segregated areas (and even of the vacuolar sap at early stages of vacuole formation) resembles the matrix of cytoplasm populated with ribosomes. The cytoplasm in the whole central area is segregated at early stages of differentiation of root cap cells. The process of autophagic vacuoles formation in which the preexisting endomembranes seemingly are involved is described. It is proposed that such vacuoles may form after separation of two unit membranes of a cisternae surrounding a portion of the cytoplasm or an organelle, followed by independent growth of an outer unit membrane.

Existence and Multiplicity of Solutions for a Robin Problem Involving the p(x)-Laplace Operator

We study the following nonlinear Robin boundary-value problem −Δp(x)u=λf(x,u) in Ω, |∇u|p(x)-2(∂u/∂v)+β(x)|u|p(x)−2u=0 on ∂Ω, where Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω, ∂u/∂v is the outer unit normal derivative on ∂Ω, λ>0 is a real number, p is a continuous function on Ω¯ with infx∈Ω¯p(x)>1, β∈L∞(∂Ω) with β−:=infx∈∂Ωβ(x)>0, and f:Ω×ℝ→ℝ is a continuous function. Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.

Skinning measures in negative curvature and equidistribution of equidistant submanifolds

Let$C$be a locally convex closed subset of a negatively curved Riemannian manifold$M$. We define the skinning measure${\sigma }_{C} $on the outer unit normal bundle to$C$in$M$by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to$C$equidistribute to the Bowen–Margulis measure${m}_{\mathrm{BM} } $on${T}^{1} M$, assuming only that${m}_{\mathrm{BM} } $is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

Study on Cluster Supply Chain Synergetic Management

Corporation with others enterprises is very important for their development, especially for Small and Mid-sized Enterprises. This paper discussed management methods among Small and Mid-sized Enterprises with synergetic thinking, based on the state of these enterprises and cluster supply chain, which provided a theoretical support for the development of small and mid-sized enterprises in China. Synergetic management in the inner unit and outer unit of cluster supply chain is described. The key safeguards to development of cluster supply chain are summarized at the end of this paper.

Multiple solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions

Using variational methods we study the non-existence and multiplicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the formwhere Ω; is a bounded domain with smooth boundary, n is the outer unit normal to ∂Ω and λ is a parameter. Furthermore, we want to emphasize that g : ∂Ω × [0,∞)→ ℝ is a continuous function that may or may not satisfy the Ambrosetti–Rabinowitz-type condition.

Lower Cambrian anabaritids from south China

Anabaritids are a distinctive group of early skeletal fossils, with tri-radially symmetrical tubes. They occur in the earliest Cambrian deposits, and seem to be especially characteristic of the Tommotian strata and putative pre-Tommotian units (e.g. Manykay horizon). A detailed redescription ofAnabarites rotundusQian from near Kuanchuanpu, Shaanxi Province emphasizes the degree of morphological variation. New data on shell composition suggest a bilayered structure, the outer unit composed of fibres parallel to growth lines, overlying a distinctive inner lamella with a chevron ultrastructure. Tentative evidence for anabaritids possibly being operculate is presented, the putative operculum being a simple convex disc. More poorly preserved anabaritids from the Yangtze Gorges (Hubei Province) and Meishucun (Yunnan Province) are referred toA. trisulcatus. All these Chinese occurrences of anabaritids are consistent with a lower Tommotian age of the associated faunas, but reliable and precise correlations with other Precambrian-Cambrian boundary sections remain inconclusive.