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

9404358加洛法罗该奖项支持对偏微分方程式领域中出现的问题进行数学研究。这项工作涉及三个主要项目。第一个问题是关于仿照Baouendi-Grushin算子的一类次椭圆算子的唯一延拓以及这类算子的嵌入本征值的不存在，或者关于Heisenberg群或一般在幂零齐次李群上的次拉普拉斯算子的嵌入本征值。第二条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形下的Sobolev泛函的Euler方程，等周和Sobolev不等式，Heisenberg群上Lipschitz区域的调和和曲面测度之间的反应，以及Kolmogorov方程的Wieners判据。我们还将讨论偏微分方程组和Pompeu问题的对称性，DeGiorgi猜想和流通过平均曲率联系在一起，Sobolev中的极值函数或Baouendi-Grushin算子或Heisenberg上的次拉普拉斯算子的等周不等式和广义扭转问题。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***