Nonsymmetric, Noncommutative, Non-Lipschitz Problems for Scale Invariant Elliptic Operators

9706648 Verchota首席调查员希望研究欧氏空间中尺度不变椭圆方程解在某些经常假设的假设的修改下的解，目的是更好地了解它们所起的作用。因此，将考虑方程组的对称性，具有有界系数的高阶算子的非对易或非变分形式，或非Lipschitz边界上的边值问题。将使用调和分析技术，Lipschitz和非Lipschitz边界上的奇异积分，以及现代椭圆型偏微分方程理论。虽然该建议停留在椭圆形偏微分方程的理论框架内，但其主题与工程、数值分析和应用数学中的应用有关。所考虑的方程的特例已经被用来模拟一段时间了，例如，固体中的电荷分布、温度分布，或者弹性物体在施加在其外表面或边界上的应力作用下可能经历的位移。这里和最近强调的比例不变性允许物体具有任意数量的角和边(这是在许多材料中自然发生的事情，例如在晶体中)。这是因为无论是通过显微镜还是通过望远镜的错误一端观察角度，角度的测量都是相同的，即这些非常基本的量是比例不变的。相比之下，我们对某些材料表面光滑程度的看法在放大后发生了巨大变化，即比例的变化。通过将我们的理论建立在比那些描述平稳性的数量更基本的数量上，并获得对建模的数量(温度等)的不同结果和估计。我们得到了一个既可以应用于边界粗糙的物体，也可以应用于光滑边界的物体的理论。这一提议可以被看作是对其他更基本的量的探索，从而导致进一步的推广或应用。

9706648 Verchota The Principle Investigator wishes to study solutions to scale invariant elliptic equations in Euclidean spaces under modification of certain frequently assumed hypotheses, with the goal of better understanding the role they play. Hence, symmetry for systems of equations, non-commutativity or nonvariational formulation for higher order operators with bounded coefficients, or boundary values on non-Lipschitz boundaries will be considered. Harmonic analysis techniques, singular integrals over Lipschitz and non-Lipschitz boundaries, and modern elliptic Partial Differential Equation theory will be used. Though the proposal stays within the theoretic framework of elliptic PDE, the subject matter is related to applications in engineering, numerical analysis, and applied mathematics. Special cases of the equations considered have been used for some time to model, for example, the distribution of electrical charges, distribution of temperatures in a solid body, or the displacements an elastic body can undergo under stresses imposed on its outside surface or boundary. The emphasis here and recently on scale invariance allows for bodies that have arbitrary numbers of corners and edges (something that occurs naturally in many materials, e.g. in crystals). This is because the measure of angles remains the same whether the angles are viewed through a microscope or through the wrong end of a telescope, i.e. these very elementary quantities are scale invariant. In contrast, our idea of how smooth the surface of some material might be changes dramatically upon magnification, i.e. change of scale. By grounding our theory in quantities more elementary than those which describe smoothness and obtaining various results and estimates on the quantities modeled (temperature, etc.) we obtain a theory that can be applied both to bodies with rough boundaries or to bodies with smooth boundaries. The proposal can be seen as a search for other more elementary quantities leading to further generalizations or applications.