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Articles de revues sur le sujet « Moving planes method »

Auteur : Grafiati

Publié le 9 mars 2023

Mis à jour le 19 juillet 2025

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1

Dancer, E. N. "Some notes on the method of moving planes." Bulletin of the Australian Mathematical Society 46, no. 3 (1992): 425–34. http://dx.doi.org/10.1017/s0004972700012089.

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In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.

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Li, Yan Yan. "Harnack Type Inequality: the Method of Moving Planes." Communications in Mathematical Physics 200, no. 2 (1999): 421–44. http://dx.doi.org/10.1007/s002200050536.

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Berestycki, H., and L. Nirenberg. "On the method of moving planes and the sliding method." Boletim da Sociedade Brasileira de Matem�tica 22, no. 1 (1991): 1–37. http://dx.doi.org/10.1007/bf01244896.

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Chen, Wenxiong, Pengyan Wang, Yahui Niu, and Yunyun Hu. "Asymptotic method of moving planes for fractional parabolic equations." Advances in Mathematics 377 (January 2021): 107463. http://dx.doi.org/10.1016/j.aim.2020.107463.

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Zhang, Lihong, and Xiaofeng Nie. "A direct method of moving planes for the Logarithmic Laplacian." Applied Mathematics Letters 118 (August 2021): 107141. http://dx.doi.org/10.1016/j.aml.2021.107141.

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Chen, Wenxiong, Congming Li, and Yan Li. "A direct method of moving planes for the fractional Laplacian." Advances in Mathematics 308 (February 2017): 404–37. http://dx.doi.org/10.1016/j.aim.2016.11.038.

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7

Lin, Chang-Shou, and Juncheng Wei. "Uniqueness of Multiple-spike Solutions via the Method of Moving Planes." Pure and Applied Mathematics Quarterly 3, no. 3 (2007): 689–735. http://dx.doi.org/10.4310/pamq.2007.v3.n3.a3.

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Guan, Pengfei, Chang-Shou Lin, and Guofang Wang. "Application of the method of moving planes to conformally invariant equations." Mathematische Zeitschrift 247, no. 1 (2004): 1–19. http://dx.doi.org/10.1007/s00209-003-0608-x.

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Дудукало, Д., D. Dudukalo, М. Чепчуров, M. Chepchurov, М. Вагнер, and M. Vagner. "METHOD FOR PRODUCING PLANES PARALLEL TO THE AXIS ROTATION AXIS ON LATHES." Bulletin of Belgorod State Technological University named after. V. G. Shukhov 4, no. 10 (2019): 142–48. http://dx.doi.org/10.34031/article_5db43fa622b135.74427811.

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This article describes the method of obtaining surfaces by moving cutting tool edge in the transversal direction on automatic lathes. The application of this handling method allows person to take a fresh look at the usage and modernization for state of the art equipment and improve its technological capabilities, which allows increasing capacity and extending the range of products. In this article tool movements are calculated for revolution of the workpiece, a model for moving the tool path is produced and with the help of using software package a tool movement graph is constructed when a plane is formed parallel to the product axis. It is established that the obtained model of tool movement allows a person to analyze velocity variation in speed from zero to maximum value, since the method’s implementation of moving the tool in the opposite direction it require to solve the tool’s reverse problem. The method allows a person to analyze movements and trajectories for ensuring the implementation of obtaining planes and various complex products on automatic lathes.

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Liu, Xingyu. "The Radial Symmetry and Monotonicity of Solutions of Fractional Parabolic Equations in the Unit Ball." Symmetry 17, no. 5 (2025): 781. https://doi.org/10.3390/sym17050781.

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We use the method of moving planes to prove the radial symmetry and monotonicity of solutions of fractional parabolic equations in the unit ball. Since the fractional Laplacian operator is a linear operator, we investigate the maximal regularity of nonlocal parabolic fractional Laplacian equations in the unit ball. The maximal regularity of nonlocal parabolic fractional Laplacian equations guarantees the existence of solutions in the unit ball. Based on these conditions, we first establish a maximum principle in a parabolic cylinder, and the principles provide a starting position to apply the method of moving planes. Then, we consider the fractional parabolic equations and derive the radial symmetry and monotonicity of solutions in the unit ball.

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Wang, Pengyan, and Pengcheng Niu. "A direct method of moving planes for a fully nonlinear nonlocal system." Communications on Pure & Applied Analysis 16, no. 5 (2017): 1707–18. http://dx.doi.org/10.3934/cpaa.2017082.

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Ying, Wang. "The Method of Moving Planes for Integral Equation in an Extremal Case." Journal of Partial Differential Equations 29, no. 3 (2016): 246–54. http://dx.doi.org/10.4208/jpde.v29.n3.6.

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Liu, Baiyu. "Direct method of moving planes for logarithmic Laplacian system in bounded domains." Discrete & Continuous Dynamical Systems - A 38, no. 10 (2018): 5339–49. http://dx.doi.org/10.3934/dcds.2018235.

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Shi, Wei. "Liouville-Type Theorem for Nonlinear Elliptic Equations Involving Generalized Greiner Operator." Mathematics 11, no. 1 (2022): 61. http://dx.doi.org/10.3390/math11010061.

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Ma, Shih-Hsin, Jun-Yi Wu, and Chun-Ming Chiang. "Drawing the Light Paths at a Lens to Find Its Effective Focal Length and Principal Planes." Physics Teacher 60, no. 7 (2022): 591–93. http://dx.doi.org/10.1119/5.0020125.

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This paper proposes a simple method to design experiments for drawing the light paths at a lens to find its effective focal length and principal planes. In the designed experiments, long-exposure photography was used to record the light scattered using a moving sheet of paper, thus revealing the light path. According to the proposed experimental method, (1) the effective focal length of a lens and (2) the principal planes of a lens can be measured. Moreover, the system is easy to build, inexpensive, and can promote students’ understanding of geometric optics by providing an intuitive physical phenomenon. The work is also of pedagogic interest as it reveals a new simple experimental method to find the principal planes of a thick lens and its effective focal length.

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Bendjilali, K., and F. Belkhouche. "Collision course by transformation of coordinates and plane decomposition." Robotica 27, no. 4 (2009): 499–509. http://dx.doi.org/10.1017/s0263574708004888.

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SUMMARYThis paper deals with the problem of collision course checking in a dynamic environment for mobile robotics applications. Our method is based on the relative kinematic equations between moving objects. These kinematic equations are written under polar form. A transformation of coordinates is derived. Under this transformation, collision between two moving objects is reduced to collision between a stationary object and a virtual moving object. In addition to the direct collision course, we define the indirect collision course, which is more critical and difficult to detect. Under this formulation, the collision course problem is simplified, and complex scenarios are reduced to simple scenarios. In three-dimensional (3D) settings, the working space is decomposed into two planes: the horizontal plane and the vertical plane. The collision course detection in 3D is studied in the vertical and horizontal planes using 2D techniques. This formulation brings important simplifications to the collision course detection problem even in the most critical and difficult scenarios. An extensive simulation is used to illustrate the method in 2D and 3D working spaces.

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Liu, Chi-Min. "Extended Stokes' Problems for Relatively Moving Porous Half-Planes." Mathematical Problems in Engineering 2009 (2009): 1–10. http://dx.doi.org/10.1155/2009/185965.

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A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

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Ciraolo, Giulio, and Luigi Vezzoni. "A sharp quantitative version of Alexandrov's theorem via the method of moving planes." Journal of the European Mathematical Society 20, no. 2 (2018): 261–99. http://dx.doi.org/10.4171/jems/766.

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Fang, Zhongbo, and Anna Wang. "Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method." Advances in Pure Mathematics 02, no. 06 (2012): 363–66. http://dx.doi.org/10.4236/apm.2012.26053.

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Pucci, C., and Colesanti Andrea. "A symmetry result for the p-laplacian equation via the moving planes method." Applicable Analysis 55, no. 3-4 (1994): 207–13. http://dx.doi.org/10.1080/00036819408840300.

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Cheng, Chunxia, Zhongxue Lü, and Yingshu Lü. "A direct method of moving planes for the system of the fractional Laplacian." Pacific Journal of Mathematics 290, no. 2 (2017): 301–20. http://dx.doi.org/10.2140/pjm.2017.290.301.

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22

Chen, Chiun-Chuan, and Chang-Shou Lin. "Estimates of the conformal scalar curvature equation via the method of moving planes." Communications on Pure and Applied Mathematics 50, no. 10 (1997): 971–1017. http://dx.doi.org/10.1002/(sici)1097-0312(199710)50:10<971::aid-cpa2>3.0.co;2-d.

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Lin, Chang-Shou. "Estimates of the scalar curvature equation via the method of moving planes III." Communications on Pure and Applied Mathematics 53, no. 5 (2000): 611–46. http://dx.doi.org/10.1002/(sici)1097-0312(200005)53:5<611::aid-cpa4>3.0.co;2-n.

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24

Liu, Xingyu. "A System of Parabolic Laplacian Equations That Are Interrelated and Radial Symmetry of Solutions." Symmetry 17, no. 7 (2025): 1112. https://doi.org/10.3390/sym17071112.

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We utilize the moving planes technique to prove the radial symmetry along with the monotonic characteristics of solutions for a system of parabolic Laplacian equations. In this system, the solutions of the two equations are interdependent, with the solution of one equation depending on the function of the other. By use of the maximal regularity theory that has been established for fractional parabolic equations, we ensure the solvability of these systems. Our initial step is to formulate a narrow region principle within a parabolic cylinder. This principle serves as a theoretical basis for implementing the moving planes method. Following this, we focus our attention on parabolic systems with fractional Laplacian equations and deduce that the solutions are radial symmetric and monotonic when restricted to the unit ball.

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Deng, Yan, Junfang Zhao, and Baozeng Chu. "Symmetry of positive solutions for systems of fractional Hartree equations." Discrete & Continuous Dynamical Systems - S 14, no. 9 (2021): 3085. http://dx.doi.org/10.3934/dcdss.2021079.

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<p style='text-indent:20px;'>In this paper, we deal with a system of fractional Hartree equations. By means of a direct method of moving planes, the radial symmetry and monotonicity of positive solutions are presented.</p>

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Mikyoung Hur, Vera. "Symmetry of steady periodic water waves with vorticity." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1858 (2007): 2203–14. http://dx.doi.org/10.1098/rsta.2007.2002.

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The symmetry and monotonicity properties of steady periodic gravity water waves are established for arbitrary vorticities if the wave profile is monotone near the trough and every streamline attains a minimum below the trough. The proof uses the method of moving planes.

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Hollifield, Elliott. "Positivity of nonnegative solutions to a system of fractional Laplacian problems in a ball." Proceedings of the American Mathematical Society, Series B 11, no. 44 (2024): 499–507. http://dx.doi.org/10.1090/bproc/240.

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We consider a cooperative system of equations involving the fractional Laplacian operator with zero Dirichlet external condition on a ball. We prove that nonnegative solutions of such problems, with semipositone nonlinearities, are positive and hence radially symmetric. We use the method of moving planes to establish our result.

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Dou, Meixia. "A direct method of moving planes for fractional Laplacian equations in the unit ball." Communications on Pure and Applied Analysis 15, no. 5 (2016): 1797–807. http://dx.doi.org/10.3934/cpaa.2016015.

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Chen, Chiun-Chuan, and Chang-Shou Lin. "Estimate of the conformal scalar curvature equation via the method of moving planes. II." Journal of Differential Geometry 49, no. 1 (1998): 115–78. http://dx.doi.org/10.4310/jdg/1214460938.

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Chen, Chiun-Chuan, and Ming Di Lee. "On the method of moving planes and symmetry of solutions of semilinear elliptic equations." Nonlinear Analysis: Theory, Methods & Applications 28, no. 10 (1997): 1697–707. http://dx.doi.org/10.1016/0362-546x(95)00240-v.

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Salehi, Nahid, and Mankyu Sung. "Realistic Multi-Agent Formation Using Discretionary Group Behavior (DGB)." Applied Sciences 10, no. 10 (2020): 3518. http://dx.doi.org/10.3390/app10103518.

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Simulating groups and their behaviors have been one of the important topics recently. This paper proposes a novel velocity-based method to simulate the realistic behavior of groups moving in a specific formation in a virtual environment including other groups and obstacles. The proposed algorithm, we called “DGB—Discretionary Group Behavior”, takes advantage of ORCA (Optimal Reciprocal Collision Avoidance) half-planes for both grouping and collision avoidance strategy. By considering new half-planes for each agent, we can have more reasonable and intelligent behavior in front of challenging obstacles and other agents. Unlike recent similar works, independent members in a group do not have predefined connections to each other even though they can keep the group’s formation while moving and trying to follow their best neighbors discretionarily in critical situations. Through experiments, we found that the proposed algorithm can yield more human-like group behavior in a crowd of agents.

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Zhang, Tao. "Liouville Type Theorems Involving the Fractional Laplacian on the Upper Half Euclidean Space." Fractal and Fractional 6, no. 12 (2022): 738. http://dx.doi.org/10.3390/fractalfract6120738.

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In this paper, we mainly establish Liouville-type theorems for the elliptic semi-linear equations involving the fractional Laplacian on the upper half of Euclidean space. We employ a direct approach by studying an equivalent integral equation instead of using the conventional extension method. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions under very weak conditions. We also extend the results to a more general equation.

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Wang, Xin-jing, and Peng-cheng Niu. "A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group." Acta Mathematicae Applicatae Sinica, English Series 37, no. 2 (2021): 364–79. http://dx.doi.org/10.1007/s10255-021-1016-x.

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Sun, Tao, and Hua Su. "Monotonicity and Symmetry of Solutions to Fractional Laplacian in Strips." Journal of Function Spaces 2021 (November 10, 2021): 1–5. http://dx.doi.org/10.1155/2021/5354775.

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In this paper, using the method of moving planes, we study the monotonicity in some directions and symmetry of the Dirichlet problem involving the fractional Laplacian − Δ α / 2 u x = f u x , x ∈ Ω , u x > 0 , x ∈ Ω , u x = 0 , x ∈ ℝ n \ Ω , in a slab-like domain Ω = ℝ n − 1 × 0 , h ⊂ ℝ n .

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Parrot, J. F., and C. Ramírez-Núñez. "POSITIVE AND NEGATIVE ROUGHNESS ACCORDING TO LOCAL DIFFERENCES BETWEEN DEM SURFACE AND 3D REFERENCE PLANES." ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences V-3-2021 (June 17, 2021): 159–65. http://dx.doi.org/10.5194/isprs-annals-v-3-2021-159-2021.

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Abstract. The irregularities of the earth’s surface are quantified by means of roughness measurements using Digital Elevation Models (DEM’s). This article presents a roughness measurement method that is based on the calculation of the difference of altitude existing between a plane passing through the centre of a moving window and the altitude of the DEM surface inside this window. This method differs from the measure of the standard deviation and best fit plane, in the sense that it considers all difference values, positives or negatives. The measurement is done in a 3 × 3 or a 5 × 5 moving window and contemplates inside this window the plane which passes through the centre of the window and the highest pixel located in the border or perimeter of this window. According to the 3D configuration of the DEM surface inside the moving window, the sum of all the differences is positive or negative, allowing to discriminate the local morphology independently of the global roughness. The roughness variable which distinguishes negative and positive values allows to classify accurately landscape units such as watersheds, riverbeds, volcanic assemblages as well as landforms associated with tectonic structures.

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Daiji, Yongzhi, Shuibo Huang, and Qiaoyu Tian. "Symmetry and Monotonicity of Solutions to Mixed Local and Nonlocal Weighted Elliptic Equations." Axioms 14, no. 2 (2025): 139. https://doi.org/10.3390/axioms14020139.

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This paper focuses on the symmetry and monotonicity of non-negative solutions to a mixed local and nonlocal weighted elliptic problem. This problem generalizes the ground-state representation of elliptic equations with the Hardy potential. The novelty of this research lies in developing the moving planes method for mixed local and nonlocal equations with a weighted function, thus clarifying the influence of the weighted function on the solution properties.

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Yang, Jonghyun, Jeongjun Park, and Changwan Yu. "Accuracy Verification of 3D Motion Analysis System Using Smart-phone Monocular Camera." Korean Journal of Sport Science 32, no. 4 (2021): 464–71. http://dx.doi.org/10.24985/kjss.2021.32.4.464.

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PURPOSE This study aimed to verify the accuracy of three-dimensional (3D) motion data produced through artificial intelligence-based user motion recognition technology with images obtained using a smartphone monocular camera. This was done to explore the possibility of developing an application that can improve the reliability of the measurement of physical activity performing motions and feedback provision.METHODS To check the accuracy of the artificial intelligence-based 3D motion analysis system that utilized a semi-supervised learning method, a commercialized 3D infrared motion analysis system measured and compared motions on three movement planes, motions with limited joint movement, and fast motions in a wide moving range.RESULTS The motions on the coronal and sagittal planes produced through the 3D motion analysis application showed very high measurement accuracy; however, the accuracy of the measurement of motions on the horizontal plane, which could not be measured directly with a camera, was relatively lower than that of the coronal and sagittal planes. Accuracy in measuring 3D motion was moderate in moving motions and low in motions with limited joint movement.CONCLUSIONS For the developed 3D motion analysis system to be used in online physical education, the types of physical activities included in the program should be comprehensively composed through the analysis of the content system of the physical education curriculum and the resultant physical activities.

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Park , Eun-Seok, Saba Arshad, and Tae-Hyoung Park. "Initial Pose Estimation Method for Robust LiDAR-Inertial Calibration and Mapping." Sensors 24, no. 24 (2024): 8199. https://doi.org/10.3390/s24248199.

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Handheld LiDAR scanners, which typically consist of a LiDAR sensor, Inertial Measurement Unit, and processor, enable data capture while moving, offering flexibility for various applications, including indoor and outdoor 3D mapping in fields such as architecture and civil engineering. Unlike fixed LiDAR systems, handheld devices allow data collection from different angles, but this mobility introduces challenges in data quality, particularly when initial calibration between sensors is not precise. Accurate LiDAR-IMU calibration, essential for mapping accuracy in Simultaneous Localization and Mapping applications, involves precise alignment of the sensors’ extrinsic parameters. This research presents a robust initial pose calibration method for LiDAR-IMU systems in handheld devices, specifically designed for indoor environments. The research contributions are twofold. Firstly, we present a robust plane detection method for LiDAR data. This plane detection method removes the noise caused by mobility of scanning device and provides accurate planes for precise LiDAR initial pose estimation. Secondly, we present a robust planes-aided LiDAR calibration method that estimates the initial pose. By employing this LiDAR calibration method, an efficient LiDAR-IMU calibration is achieved for accurate mapping. Experimental results demonstrate that the proposed method achieves lower calibration errors and improved computational efficiency compared to existing methods.

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Dai, Wei, Yanqin Fang, and Guolin Qin. "Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes." Journal of Differential Equations 265, no. 5 (2018): 2044–63. http://dx.doi.org/10.1016/j.jde.2018.04.026.

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张, 晓亚. "The Method of Moving Planes for Fractional Order Elliptic Equations with Hardy and General Nonlinear Terms." Advances in Applied Mathematics 12, no. 09 (2023): 3804–13. http://dx.doi.org/10.12677/aam.2023.129374.

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Hou, Wenwen, Lihong Zhang, Ravi P. Agarwal, and Guotao Wang. "Radial symmetry for a generalized nonlinear fractional p-Laplacian problem." Nonlinear Analysis: Modelling and Control 26, no. 2 (2021): 349–62. http://dx.doi.org/10.15388/namc.2021.26.22358.

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This paper first introduces a generalized fractional p-Laplacian operator (–Δ)sF;p. By using the direct method of moving planes, with the help of two lemmas, namely decay at infinity and narrow region principle involving the generalized fractional p-Laplacian, we study the monotonicity and radial symmetry of positive solutions of a generalized fractional p-Laplacian equation with negative power. In addition, a similar conclusion is also given for a generalized Hénon-type nonlinear fractional p-Laplacian equation.

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MOTA, Rafael, Ray KIRBY, Paul WILLIAMS, and Stefan JACOB. "Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 270, no. 9 (2024): 2711–18. http://dx.doi.org/10.3397/in_2024_3223.

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Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

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Tang, Sufang, and Jingbo Dou. "Nonexistence results for a fractional Hénon–Lane–Emden equation on a half-space." International Journal of Mathematics 26, no. 13 (2015): 1550110. http://dx.doi.org/10.1142/s0129167x15501104.

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Consider the following Dirichlet problem involving the fractional Hénon–Lane–Emden Laplacian: [Formula: see text] where [Formula: see text] and [Formula: see text] is the upper half-Euclidean space. We first show that the above equation is equivalent to the following integral equation: [Formula: see text] where [Formula: see text] is the Green function in [Formula: see text] with the same Dirichlet condition. Then we prove the nonexistence of positive solutions by using the method of moving planes in integral forms.

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Hou, Wenwen, and Lihong Zhang. "Radial symmetry of a relativistic Schrödinger tempered fractional p-Laplacian model with logarithmic nonlinearity." Nonlinear Analysis: Modelling and Control 28 (December 23, 2022): 1–14. http://dx.doi.org/10.15388/namc.2023.28.29621.

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In this paper, by introducing a relativistic Schrödinger tempered fractional p-Laplacian operator (–Δ)p,λs,m, based on the relativistic Schrödinger operator (–Δ + m2)s and the tempered fractional Laplacian (Δ + λ)β/2, we consider a relativistic Schrödinger tempered fractional p-Laplacian model involving logarithmic nonlinearity. We first establish maximum principle and boundary estimate, which play a very crucial role in the later process. Then we obtain radial symmetry and monotonicity results by using the direct method of moving planes.

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Zhang, Tao, and Tingzhi Cheng. "A priori estimates of solutions to nonlinear fractional Laplacian equation." Electronic Research Archive 31, no. 2 (2022): 1119–33. http://dx.doi.org/10.3934/era.2023056.

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<abstract><p>In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.</p></abstract>

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Akmatov, A. "The Regularization Method of Solutions a Bisingularly Perturbed Problem in the Generalized Functions Space." Bulletin of Science and Practice 8, no. 2 (2022): 10–17. http://dx.doi.org/10.33619/2414-2948/75/01.

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When singularly perturbed problems are investigated, in the case of a change in stability, all work was performed in the space of analytical functions. Naturally, questions will arise whether it is possible to obtain an estimate of solutions to a singularly perturbed problem without moving to the complex plane. In the work, the first results obtained are the solutions of the singularly motivated task, not moving into the complex plane. For this purpose, a method of regularization in the space of generalized functions has been developed and corresponding estimates have been obtained. If we choose the starting point in a stable interval, then up to the transition point, the asymptotic proximity of solutions to the perturbed and undisturbed problem is in the order of a small parameter ε. The problem will appear when the point belongs to an unstable interval. Therefore, prior to this, the works moved to the complex plane. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. At special points, these lines have critical level lines. Therefore, it is impossible to choose the starting point so as to get the maximum delay time. But the asymptotic proximity of solutions of perturbed and undisturbed problems is possible with limited time delays. If we study the solution in the space of generalized functions, then we can choose the starting point with the maximum time delay. And also, without passing to the complex plane, it is possible to establish the asymptotic proximity of solutions to the perturbed and undisturbed problem. For this purpose, a method of regularization of solutions of a singularly perturbed problem has been developed for the first time.

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FELMER, PATRICIO, and YING WANG. "RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN." Communications in Contemporary Mathematics 16, no. 01 (2014): 1350023. http://dx.doi.org/10.1142/s0219199713500235.

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The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem [Formula: see text] where (-Δ)αdenotes the fractional Laplacian, α ∈ (0, 1), and B1denotes the open unit ball centered at the origin in ℝNwith N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1→ ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation [Formula: see text] with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing.Our third goal is to consider radial symmetry of positive solutions for system of the form [Formula: see text] where α1, α2∈(0, 1), the functions f1and f2are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1and g2are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

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Shiller, Zvi, and William Serate. "Trajectory Planning of Tracked Vehicles." Journal of Dynamic Systems, Measurement, and Control 117, no. 4 (1995): 619–24. http://dx.doi.org/10.1115/1.2801122.

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This paper presents a method for computing the track forces and track speeds of planar tracked vehicles, required to follow a given path at specified speeds on horizontal and inclined planes. It is shown that the motions of a planar tracked vehicle are constrained by a velocity dependent nonholonomic constraint, derived from the force equation perpendicular to the tracks. This reduces the trajectory planning problem to determining the slip angle between the vehicle and the path tangent that satisfies the nonholonomic constraint along the entire path. Once the slip angle has been determined, the track forces are computed from the remaining equations of motion. Computing the slip angle is shown to be an initial boundary-value problem, formulated as a parameter optimization. This computational scheme is demonstrated numerically for a planar vehicle moving along circular paths on horizontal and inclined planes.

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Le, Phuong, and Hoang-Hung Vo. "Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips." Communications on Pure & Applied Analysis 21, no. 3 (2022): 1027. http://dx.doi.org/10.3934/cpaa.2022008.

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<p style='text-indent:20px;'>By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [<xref ref-type="bibr" rid="b16">16</xref>,<xref ref-type="bibr" rid="b20">20</xref>] to the system, in which substantial differences with the single cases are presented.</p>

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Kang, Hyeonbae, and Shigeru Sakaguchi. "A symmetry theorem in two-phase heat conductors." Mathematics in Engineering 5, no. 3 (2022): 1–7. http://dx.doi.org/10.3934/mine.2023061.

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<abstract><p>We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $ C^{2, \alpha} $, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.</p></abstract>

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