- Title Information
- Title
- Elliptic Boundary Value Problems on Irregular Domains
- Name:
Personal
- Name Part
- Li, Zongyuan
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Dong, Hongjie
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Guo, Yan
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Dafermos, Constantine
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2020
- Physical Description
- Extent
- viii, 196 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2020
- Genre (aat)
- theses
- Abstract
- In 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 $q\in[1,(m+2)/(m+1))$.
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01012163")
- Topic
- Mathematics
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20200720
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations