Artur Andrade, Temple University
Abstract: Elliptic boundary value problems arise naturally in modeling a wide range of physical phenomena, including electrostatics, elasticity, steady-state incompressible fluid flow, and electromagnetism. A powerful tool for the treatment of such problems is the layer potential method, through which matters are reduced to solving a boundary integral equation involving a singular integral operator naturally associated with the domain, and a coefficient tensor for the underlying PDE. When this singular integral operator is compact, the boundary integral equation can be treated using Fredholm Theory. While systematic progress has been made in the study of second-order elliptic systems along these lines, the case of higher-order elliptic systems remains far less understood.
In this talk, I will present a distinguished coefficient tensor for the polyharmonic operator $\Delta^3$ in all dimensions, and illustrate how the associated singular integral operator is compact on $L^p$ Lebesgue-type spaces, for all integrability exponents $p\in(1,\infty)$, thus opening the door for the employment of Fredholm Theory for the solvability of the Dirichlet Problem for $\Delta^3$ in infinitesimally flat AR domains.
This is an ongoing work with Dorina Mitrea (Baylor University), Irina Mitrea (Temple University), and Marius Mitrea (Baylor University).