项目名称: 一维平均曲率方程定性问题研究
项目编号: No.11201008
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 王立波
作者单位: 北华大学
项目金额: 22万元
中文摘要: 平均曲率方程来源于几何和物理,在流体力学、毛细作用理论、限流扩散现象、微机电系统中都有广泛的应用。本项目关注于一维平均曲率方程的定性性质,包括: 1)利用非光滑临界点理论、变分不等式理论等,讨论曲率摆方程周期解的存在性问题; 2) 利用分析方法、时间映射、比较原理结合整体分歧思想等研究一维平均曲率方程典型问题的精确解的个数、边界爆破解及正则性、渐近性等; 3) 利用非光滑临界点理论、变分不等式理论,Leray-Schauder度理论、上下解方法、能量估计等,对比研究微机电系统模型的古典解与非古典解,获得正解、爆破解、基态解的存在性结果,并讨论解的正则性。 通过对一维平均曲率方程完整、细致的研究,一方面完善并丰富一维平均曲率方程的研究成果;另一方面,为更复杂的高维情形的研究提供借鉴和帮助。
中文关键词: 平均曲率方程;微机电系统;定性性质;径向解;混合单调迭代
英文摘要: Mean curvature equations arise in geometry and physics,but also in fluid mechanics,capillarity theory, flux limited diffusion phenomena and micro-electr- omechanical systems. The project plan to study qualitative property of one dimensional mean curvature equation systematically, include 1)using non-smooth critical point theory and variational inequality theory, we consider the periodic solution of forced relativistic pendulum equations; 2) combing with analysis technique, time map and global bifurcation theory, we consider the number of exact solutions, boundary blowup solution, regularity and asymptotic behavior of the one dimensional mean curvature equation, where the right nonlinearity is the typical functions; 3) applying non-smooth critical point theory, variational inequalities, Leray-Schauder degree, lower and upper solutions and energy estimate, we comparative study the classical and non-classical solutions of the micro-electro mechanical systems, and obtain the existence and multiplicity of positive solution, blowup solutions, ground state solutions, and the regularity of solutions. We believe that a complete description for one-dimensional mean curvature equations can not only improve and enrich the research of one-dimensional case, but also be helpful to explore more complicated high-dimensional prob
英文关键词: Mean curvature equations;Micro-electromechanical systems;Qualitative property;Radial solutions;Mixed monotone iterative