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

9404358 Garofalo该奖项支持对偏微分方程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。第一个问题是关于一类以baouendigrushin算子为模型的亚椭圆算子的唯一延拓，以及此类算子的内嵌特征值的缺失，或者是海森堡群上的次拉普拉斯算子，或者一般来说，幂零齐次李群上的子拉普拉斯算子的唯一延拓。第二部分将研究非二次情况下Sobolev泛函的Euler方程、等周不等式和Sobolev不等式、Heisenberg群上Lipschitz域的调和与表面测度之间的反应以及Kolmogorov方程的wiener准则等非线性次椭圆方程解的最优正则性。我们还将研究偏微分方程的对称性和Pompieu问题，关于DeGiorgi猜想与平均曲率相关的流动，Sobolev中的极值函数或Baouendi-Grushin算子的等周不等式或海森堡的次拉普拉斯算子和广义扭转问题。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***