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

9706648 Verchota首席研究员希望研究欧几里得空间中尺度不变椭圆方程在某些经常假设的修正下的解，目的是更好地理解它们所起的作用。因此，方程系统的对称性，具有有界系数的高阶算子的非交换性或非变分公式，或非lipschitz边界上的边值将被考虑。谐波分析技术，在Lipschitz和非Lipschitz边界上的奇异积分，以及现代椭圆偏微分方程理论将被使用。虽然该建议停留在椭圆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。所考虑的方程的特殊情况已经用了一段时间来模拟，例如，电荷的分布，固体中的温度分布，或者弹性体在施加在其外表面或边界上的应力下可以承受的位移。这里和最近对尺度不变性的强调允许物体具有任意数量的角和边（在许多材料中自然发生的事情，例如晶体）。这是因为无论从显微镜还是望远镜的另一端观察角度，角度的测量都是相同的，也就是说，这些非常基本的量是尺度不变的。相反，我们对某些材料表面的光滑程度的看法在放大后会发生巨大的变化，即尺度的变化。通过将我们的理论建立在比描述光滑性更基本的量的基础上，并根据所模拟的量（温度等）获得各种结果和估计，我们得到了一个既可以应用于边界粗糙的物体，也可以应用于边界光滑的物体的理论。这个建议可以被看作是对其他更基本量的探索，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.