Elliptic Boundary Value Problems on Irregular DomainsLi, ZongyuancreatorDong, HongjieAdvisorGuo, YanReaderDafermos, ConstantineReaderBrown University. Department of Applied Mathematicssponsor2020viii, 196 p.born digitalThesis (Ph. D.)--Brown University, 2020thesesIn this thesis, we study the elliptic boundary value problems on irregular domains. We first obtain the $W^{2,p}$ and $C^2$ regularity theories for second-order, non-divergence form elliptic equations with oblique derivative boundary conditions. In both cases, the boundary smoothness requirements are relaxed by one derivative. In particular, for the $W^{2,p}$ theory, the boundary is allowed to be Lipschitz with small constant, instead of the $C^{1,1}$ condition in the classical theory. For the $C^2$ theory, the boundary requirement is relaxed from $C^{2,Dini}$ to $C^{1,Dini}$. The second topic is the optimal regularity of second-order, divergence form elliptic equations with mixed Dirichlet-conormal boundary conditions. The aim is to find the minimum assumptions on the boundary and the interfacial boundary between the two boundary conditions, such that the ``optimal regularity'' is achieved. For this, we first develop the approximation construction to deal with locally flat domains with locally flat interfacial boundaries. Such local flatness is commonly called ``Reifenberg flat'' in the literature. Based on the construction, we obtain the optimal $W^{1,4-\epsilon}$ regularity of solutions with homogeneous boundary conditions. In a subsequent work, we further generalize our method to deal with interfacial boundaries which are locally close to Lipschitz graphs in $m$ variables with respect to the Hausdorff distance, where $m=0,\cdots,d-2$. In this direction, we obtain the optimal $W^{1,2(m+2)/(m+1)-\epsilon}$ regularity. Furthermore, when the domain is also Lipschitz, we obtain the unique solvability of the Laplace equation with $L^q+W^{1,q}$ boundary data, where $q\in[1,(m+2)/(m+1))$.MathematicsEnglishRPB20200720In CopyrightCollection is open for research.dissertations