Max Engelstein (University of Chicago)A Two-Phase Free Boundary Problem for Harmonic MeasureWe study the regularity of the free boundary for the following two-phase problem: let $\omega^+$ be the harmonic measure of a domain $\Omega$ and $\omega^-$ be the harmonic measure of $\Omega^- = (\overline{\Omega})^c$. Assume these measures are mutually absolutely continuous and let $h$, the Radon-Nikodym derivative, satisfy $\log(h) \in C^{0,\alpha}(\partial \Omega)$. We prove, under minimal geometric assumptions, that $\Omega$ is a $C^{1,\alpha}$ domain. The situation where $\log(h)$ has higher regularity is also discussed. Rohit Jain (University of Texas at Austin) Some Regularity Results for Obstacle Problems Exhibiting Non-Local BehaviorStarting from an optimal cash management problem and its formulation as a stochastic impulse control problem we will derive an obstacle problem where the obstacle exhibits non-local behavior. The goal of this presentation is to present some regularity results as well as discuss the motivating problem and comment on future work. We start with the 1d model for intuition and then proceed to discuss the higher dimensional analogue. We present two different models for analysis and develop the necessary theory from Elliptic Partial Differential Equations. We also discuss other applications arising from variants of the model particularly in Mathematical Finance and Network Theory. Lami Kim (Hokkaido University)α-Gauss curvature flows with flat sidesIn this paper, we study the deformation of the 2-dimensional convex surfaces in \mathbb R ^{3} whose speed at a point on the surface is proportional to \alpha-power of positive part of Gauss curvature. First, for \frac{1}{2}<\alpha \leq 1, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with C^{1,1}-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 < t < T_0 under certain necessary regularity and non-degeneracy initial conditions, where T_0 is the vanishing time of the flat side. Sung Sic Yoo (Inha University)Asymmetric nozzle jet flow impinging on a flat wallWe consider a two-dimensional asymmetric nozzle jet flow impinging on a flat wall. The flow issuing from nozzle orifice makes free boundaries between fluid and surrounding air. Although the governing equation is harmonic, the free boundaries make the problem highly non-linear. We propose an efficient algorithm to find the free boundaries of the asymmetric jet emanating from arbitrary nozzle. Using the boundary integral equations, our method can find the gradient toward the solution from the assumed free boundaries. Particularly, by considering the far-field behavior of solution, our algorithm can deal with the entire computational domain instead of truncated one. In the numerical results, the force exerted on the wall by the pressure and numerical calculations of diverse examples are shown.