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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Joe W. Jenkins
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Syracuse University
United States
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