This dissertation comes in two parts. The problems considered are all related to asymptotic behavior of solutions to nonlinear PDE in various scaling limits. Part I considers long time behavior for a dynamic model and Part II considers the macroscopic averaging behavior of stationary models with multiple length scales. These kinds of problems face significant additional difficulties when the interior equations are coupled with a moving boundary or irregular boundary data on a fixed boundary. This dissertation addresses several of these questions and develops new techniques to deal with PDE problems involving irregularity on a lower dimensional interface.

Chapter 2 considers a model for the contact angle motion of quasi-static capillary drops resting on a flat surface. We prove the nonlinear dynamic stability of the round drop stationary solution under a strong star-shapedness assumption on the initial data. The key tool is a new geometric a priori estimate for the regularity of the contact line.

Chapter 3 examines the case of a nonlinear elliptic equation with a periodic oscillating Dirichlet boundary condition in a general bounded domain. We prove a qualitative homogenization result under quite general conditions. Here the major difficulty is the lack of continuity of the homogenized boundary condition caused by singular behavior near boundary points with rational normal direction.

Chapter 4 is concerned with the fine properties of the homogenized boundary condition for the problem of Chapter 3. We show that the behavior depends essentially on the form of the operator. Linear operators and rotationally invariant nonlinear operators lead to a continuous homogenized boundary condition. On the other hand, for a typical nonlinear non-rotation invariant operator the homogenized boundary condition is discontinuous at every rational direction.

Finally, in Chapter 5, we discuss both the Dirichlet and Neumann boundary homogenization problems for nonlinear operators when the boundary data is a random field. The main result is quantitative homogenization in half-spaces and general domains when the boundary data has the form of an i.i.d. random checkerboard and the interior operator is non-random. The result extends as well to the strongly mixing setting with sufficient decay.