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

小行星9404358 该奖项支持在偏微分方程领域出现的问题的数学研究。 这项工作涉及三个主要项目。 第一个问题是一类以Baouendi-Grushin算子为模型的次椭圆算子的唯一延拓，以及此类算子没有嵌入本征值，或者海森堡群上的次拉普拉斯算子，或者一般而言，幂零齐次李群上的次拉普拉斯算子。 第二行的调查将研究最佳的正则性的解决方案的非线性次椭圆方程所产生的欧拉方程的Sobolev功能在非二次的情况下，等周和Sobolev不等式，调和之间的反应和表面措施的Lipschitz域上的海森堡群和Wieners标准的Kolmogorov方程。 工作也将做对称的偏微分方程和Pompieu问题，一个猜想DeGiorgi连接流的平均曲率，极值函数在索伯列夫或等周不等式的Baouendi-Grushin运营商或次拉普拉斯海森堡和广义扭转问题。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

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. ***