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Let Ω be a bounded domain in \begin{document}$\mathbb{R}^2 $\end{document} with smooth boundary, we study the following Neumann boundary value problem \begin{document}$\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$ \end{document} where \begin{document} $ν$ \end{document} denotes the outer unit normal vector to \begin{document} $\partial \Omega$ \end{document} , \begin{document} $h∈ C^{0,α}(\partial \Omega)$ \end{document} , \begin{document} $s>0$ \end{document} is a large parameter and \begin{document} $\phi_1$ \end{document} is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of \begin{document} $\phi_1$ \end{document} on the boundary as \begin{document} $s\to+∞$ \end{document} .

This work investigates a nonlocal sinh–Gordon equation with a singularly perturbed parameter in a ball. Under the Robin boundary condition, the solution asymptotically forms a quite steep boundary layer in a thin annular region, and rapidly becomes a flat curve outside this region. Focusing more particularly on the structure of the thin annular layer in this region, the pointwise asymptotic expansion involving the domain-size is evaluated more sharply, where the domain-size exactly appears in the second term of the asymptotic expansion. It should be stressed that the standard argument of matching asymptotic expansions is limited because the model has a nonlocal coefficient depending on the unknown solution. A new approach relies on integrating ideas based on a Dirichlet-to-Neumann map in an asymptotic framework. The rigorous asymptotic expansions for the thin layer structure also matches well with the numerical results. Furthermore, various boundary concentration phenomena of the thin annular layer are precisely demonstrated.

This paper is motivated by a gauged Schrodinger equation in dimension 2 including the so-called Chern–Simons term. The radially symmetric case leads to an elliptic problem with a nonlocal defocusing term, in competition with a local focusing nonlinearity. In this work we pose the equations in a ball under homogeneous Dirichlet boundary conditions. By using singular perturbation arguments we prove existence of solutions for large values of the radius. Those solutions are located close to the boundary and the limit profile is given.

Робота присвячена експериментальному дослідженню залежності електричного опору, коефіцієнту теплопровідності та пористості електропровідних композиційних матеріалів (ЕКМ) системи «кераміка-вуглець» від вмісту нанорозмірних вуглецевих наповнювачів та технологічних режимів виготовлення. Проаналізовано аналітичні залежності питомого електричного опору від деформованого стану ЕКМ у рамках теорії перколяції. Результати експериментальних досліджень показали, що зменшення дисперсності частинок електропровідної фази композиту (використання нанопластинок графіту замість термічно розширеного графіту) дозволяє значно зменшити гістерезис кривих залежності електричного опору від температури при нагріванні та охолодженні. Експериментально підтверджено вплив зміни об’єму діелектричної складової ЕКМ на електричний опір при зміні температури в межах 20–620 °C., The work is devoted of the experimental study the dependence of electrical resistance, the thermal conductivity coefficient and porosity of the electro conductive composite materials (ECM) system «ceramics-carbon» on the content of nanosized carbon fillers and technological regimes of manufacture. The analytical dependences of the specific electric resistance on the deformed state of the ECM in the percolation theory has been analyzed. The results The results of experimental studies have shown that the decrease of the dispersion of the particles of the conductive phase of the composite (usage of graphite nanoplates instead of thermally extended graphite) was able to significantly reduced the hysteresis of the curves of the resistance of the electric resistance to the temperature when heated and cooled. The influence of the change in the volume of the dielectric component of the ECM on the electrical resistance at a temperature change of 20-620 °C has been experimentally confirmed., Работа посвящена экспериментальному исследованию зависимости электрического сопротивления, коэффициента теплопроводности и пористости, электропроводящих композиционных материалов (ЭКМ) системы «керамика-углерод» от содержания наноразмерных углеродных наполнителей и технологических режимов изготовления. Проанализированы аналитические зависимости удельного электрического сопротивления от деформированного состояния ЭКМ в рамках теории перколяции. Результаты экспериментальных исследований показали, что уменьшение дисперсности частиц электропроводящей фазы композита (использование нанопластинок графита вместо термически расширенного графита) позволяет значительно уменьшить гистерезис кривых зависимости электрического сопротивления от температуры при нагревании и охлаждении. Экспериментально подтверждено влияние изменения объема диэлектрической составляющей ЭКМ на электрическое сопротивление при изменении температуры в пределах 20–620 °С.

We study the existence of steady states to the Keller–Segel system with linear chemotactical sensitivity function on a smooth bounded domain in \(\mathbb {R}^N,\)\(N\ge 3,\) having rotational symmetry. We find three different types of chemoattractant concentration which concentrate along suitable \((N-2)\)-dimensional minimal submanifolds of the boundary. The corresponding density of the cellular slime molds exhibit in the limit one or more Dirac measures supported on those boundary submanifolds.

We consider the boundary value problem { − Δ u + u = λ e u , in Ω ∂ ν u = 0 on ∂ Ω where Ω is a bounded smooth domain in R 2 , λ > 0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis. We establish the existence of a solution u λ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ → 0 . These solutions have large mass in the sense that ∫ Ω λ e u λ ∼ | log ⁡ λ | .

Several grain sizes were obtained by heat treatment at different temperatures for interstitial-free (IF) and 2.25Cr-1Mo steels. Samples of the steels with different grain sizes were aged at 600 and 680 °C for IF steel and 520 and 560 °C for 2.25Cr-1Mo steel for sufficient time to achieve their equilibrium grain boundary segregation. The grain boundary concentrations of phosphorus were examined using Auger electron spectroscopy. At the same aging temperature, the boundary segregation of phosphorus increased with increasing grain size. The effect of grain size on equilibrium grain boundary segregation thermodynamics was analyzed based on the information of both grain size and phosphorus boundary concentration. The segregation enthalpy increased with increasing grain size and simultaneously the segregation entropy became less negative. Moreover, the segregation entropy (∆S) and enthalpy (∆H) of phosphorus in both IF and 2.25Cr-1Mo steels exhibited a unified linear relationship, being expressed as ∆S = 0.85∆H − 38.06, although it segregated to different types of grain boundaries (ferrite grain boundaries in IF steel and prior austenite grain boundaries in 2.25Cr-1Mo steel). With the aid of the acquired thermodynamic parameters and grain boundary segregation theories, the equilibrium segregation concentrations at different aging temperatures were modeled under different grain sizes for both steels.

The diffusion problems with moving boundary are formulated into a general integral form. Fundamental Green's function is used to derive a transcendental equation that gives readily the interface advancement, the concentration profile, and the boundary concentration of a growing sphere. Examples dealing with the diffusion-controlled spherical growth in finite and infinite regions are calculated and compared to the results available in the literature. Potential applications of this technique are briefly discussed.