Singular Integral Operators for Higher-Order Systems in Non-Smooth Domains

Temple University, Philadelphia PA

Investigators

Abstract

The vast majority of physical phenomena are described in the language of partial differential equations. When coupled with measurements on the interface separating the region where these phenomena take place from the rest of the space, one arrives at what is commonly called a boundary value problem for the phenomenon in question. One of the most powerful techniques employed in the treatment of such boundary value problems is the method of layer potentials. This has had tremendous success both at theoretical and numerical levels for second order operators such as phenomena involving gravitational forces, heat at equilibrium, phenomena involving elastic deformations, acoustic and electromagnetic scattering, and wave propagation. By way of contrast, considerable less is presently understood for phenomena involving higher-order systems such as beam bending, plate vibrations and elastic plate deflections. The overall goal of this project is to address this issue by developing a systematic treatment of higher-order elliptic boundary value problems using singular integral operators, in a very general class of non-smooth domains. The emphasis on non-smooth structures is crucial for the viability of the theory for practical applications as physical domains exhibit asperities and irregularities of a very intricate nature. In turn, these significantly affect the properties of solutions of the partial differential equation problems. The successful completion of this project is expected to have significant impact in theoretical and applied mathematics, numerical methods, mathematical physics, and engineering, by establishing a theoretical framework which is just as effective as the traditional technologies dealing with simpler models. The project is for research in the areas of Harmonic Analysis, Geometric Measure Theory and Partial Differential Equations, and its overall aim is to develop a systematic treatment of higher-order elliptic boundary value problems using singular integral operators, in a very general class of non-smooth domains, which is in the nature of best possible from the geometric measure theoretic point of view. The main objectives of the current project are to identify: (1) the fullest family of multi-layer potential operators associated with a given homogeneous constant coefficient higher-order elliptic system; (2) geometric measure theoretic settings for which boundary multi-layer potential operators of double and single type are bounded on appropriate spaces of Whitney arrays (function spaces on the boundary suitably adapted to the higher-order setting); and (3) algebraic and geometric measure theoretic settings for which boundary multi-layer potential operators are invertible on appropriate spaces of Whitney arrays. The tools are rooted in Harmonic Analysis, Geometric Measure Theory, Partial Differential Equations, and Functional Analysis. A key step is to develop a new generation of Calderon-Zygmund theory for multi-layers acting on Whitney arrays, starting with the case when the domain is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that the domain is uniformly rectifiable. There are inherent difficulties in carrying out this program, such as lack of uniqueness of Green?s formula, algebraic difficulties, the necessity of developing a suitable function space theory (including trace, extension, and interpolation theory) for function spaces of Whitney arrays in uniformly rectifiable domains, the failure in the non-Lipschitz context of a number of key ingredients in the theory of singular integrals on Lipschitz domains such as Rellich-type estimates. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.