Mathematical Sciences: Unique Continuation, Regularity of Solutions to Linear and Nonlinear Equations of Nonelliptic Type, Symmetry for PDE's

数学科学：非椭圆型线性和非线性方程解的唯一连续性、正则性、偏微分方程的对称性

• 批准号: 9404358
• 负责人: Nicola Garofalo
• 金额: $ 14.25万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404358&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Unique; Continuation; Regularity

9404358 Garofalo This award supports mathematical research on problems arising in the field of partial differential equations. The work is concerned with three main projects. The first concerns unique continuation for a class of subelliptic operators modelled on the Baouendi-Grushin operator and absence of embedded eigenvalues for such operators, or for the sub-Laplacian on the Heisenberg group or, in general, on a nilpotent homogeneous Lie group. The second line of investigation will study optimal regularity of solutions of nonlinear subelliptic equations which arise as Euler equations of the Sobolev functional in the nonquadratic case, isoperimetric and Sobolev inequalities, reactions between harmonic and surface measure for Lipschitz domains on the Heisenberg group and Wieners criterion of Kolmogorov's equation. Work will also be done on symmetry for partial differential equations and the Pompieu problem, on a conjecture of DeGiorgi connected with flow by mean curvature, extremal functions in the Sobolev or the isoperimetric inequality for the Baouendi-Grushin operator or the sub-Laplacian on the Heisenberg and the generalized torsion problem. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

9404358 Garofalo 该奖项支持对偏微分方程领域中出现的问题的数学研究。 这项工作涉及三个主要项目。 第一个涉及以Baouendi-Grushin算子为模型的一类次椭圆算子的唯一延拓，并且此类算子缺乏嵌入特征值，或者海森堡群上的亚拉普拉斯算子，或者一般而言，幂零齐次李群上的次拉普拉斯算子。 第二条研究路线将研究非线性亚椭圆方程解的最优正则性，这些方程是非二次情况下 Sobolev 泛函的欧拉方程、等周和 Sobolev 不等式、海森堡群上 Lipschitz 域的调和与表面测度之间的反应以及维纳准则柯尔莫哥洛夫方程。 还将研究偏微分方程的对称性和庞皮欧问题，研究通过平均曲率与流动相关的德吉奥尔吉猜想、Sobolev 中的极值函数或 Bauuendi-Grushin 算子或亚拉普拉斯算子的等周不等式。海森堡和广义扭转问题。 偏微分方程构成了物理世界数学建模的基础。 数学分析的作用与其说是创建方程，不如说是提供有关解决方案的定性和定量信息。 这可能包括有关独特性、平滑性和成长性问题的答案。 此外，分析通常会开发近似解的方法并估计这些近似的准确性。 ***