Title: A uniqueness result for a p-Laplacian Dirichlet problem
Abstract: We present a uniqueness for positive solutions to a p-Laplacian Dirichlet problem when the forcing term in the equation is non-monotone and allowed to be singular at the origin.

Title: Optimal Control in Fluid Flows through Deformable Porous Media
Abstract: We consider an optimal control problem subject to an elliptic-parabolic coupled system of partial differential equations that describes fluid flow through biological tissues. Our goal is to optimize the fluid pressure and solid displacement using distributed or boundary control. We first show existence and uniqueness of an optimal control and then present necessary optimality conditions. The optimal controls can be approximated numerically.

Title: A Fredholm Alternative for elliptic equations with interior and boundary nonlinear reactions
Abstract: This talk treats two-parameter problems for a triple (a, b,m) of continuous, symmetric bilinear forms on a real separable Hilbert space V that are used for large classes of elliptic PDEs with nontrivial boundary conditions. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. The Steklov-Robin Fredholm equation is used to exemplify the abstract results.

Title: Dual-Wind Discontinuous Galerkin Method for a Parabolic Obstacle Problem and an Optimal Control Problem
Abstract: In this talk, a dual-wind discontinuous Galerkin method (DWDG) and its application to a parabolic obstacle problem and an optimal control problem (OCP) is discussed. \\ A fully discrete scheme to solve the parabolic obstacle problem with a general obstacle function in $\mathbb{R}^2$ that uses a symmetric dual-wind discontinuous Galerkin discretization in space and a backward Euler discretization in time is proposed and analyzed. The convergence of numerical solutions in $L^\infty(L^2)$ and $L^2(H^1)$ like energy norms is established and the rates are computed. Next, the DWDG method is used to discretize an infinite dimensional linear elliptic PDE constrained optimal control problem with control constraints to a finite dimensional optimization problem. The finite dimensional optimization problem is then solved with a primal-dual active set strategy to find the numerical solution $(\overline{y_h},\overline{u_h})$ which is an optimal state and control pair. Several numerical tests are provided to demonstrate the robustness and effectiveness of the proposed methods to both problems. This is a joint work with Tom Lewis, Aaron Rapp and Yi Zhang.