HARMONIC ANALYSIS METHODS FOR ELLIPTIC BOUNDARY VALUE PROBLEMS IN UNIFORMLY RECTIFIABLE AND INFINITESIMALLY FLAT AHLFORS REGULAR DOMAINS

The work in this Ph.D. thesis lies at the intersection of Harmonic Analysis, Partial Differential Equations, and Geometric Measure Theory. It focuses on the study of boundary integral operators associated with elliptic boundary value problems on non-smooth domains via singular integral methods. The results are joint work with Dorina Mitrea (Baylor University), Irina Mitrea, and Marius Mitrea (Baylor University) (cf. [2], [3], and [4]), and the thesis is structured around four interconnected themes: A. Overdetermined boundary value problems for the Laplacian on uniformly rectifiable domains; B. Overdetermined boundary value problems for second-order, homogeneous, constant complex coefficient elliptic systems on uniformly rectifiable domains; C. Overdetermined boundary value problems for the Stokes system of linearized hydrostatics on uniformly rectifiable domains; D. Fredholm Theory for the Dirichlet boundary value problem for $\Delta^{3}$ on infinitesimally flat Ahlfors regular domains in $\mathbb{R}^{n}$. For themes A and B, this dissertation contains a well-posedness theory for overdetermined boundary value problems in bounded uniformly rectifiable domains with boundary data in $L^p$-based function spaces. A key ingredient to obtaining such results is introducing a Cauchy-like singular integral operator adapted to second-order homogeneous elliptic systems and the overdetermined setting. See also [2]. For theme C, the well-posedness theory for the overdetermined boundary problem for the Stokes system of linearized hydrostatics requires the introduction of a new $L^p$-based {\it divergence-free} Lebesgue-Whitney function space, along with a pair of Cauchy-like integral operators specifically adapted to the Stokes system. Additionally, this work contains a connection between the Cauchy integrals for the Stokes system of linearized hydrostatics and those for the Lamé system of linearized elastostatics as one of the Lamé parameters tends to infinity. See also [3]. Finally, for theme D, this dissertation contains a {\it distinguished} coefficient tensor for the polyharmonic operator $\Delta^3$ in all dimensions of the Euclidean space. It also provides an argument showing that the associated singular integral operator for the Dirichlet problem for $\Delta^3$ in infinitesimally flat Ahlfors regular domains is compact on $L^p$ Lebesgue-Whitney function spaces for all $p\in(1,\infty)$, thus opening the door for the employment of Fredholm Theory for the solvability of the Dirichlet Problem. All developments presented are from ongoing joint in [4].