Title: A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$
Abstract: Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locally prescribed data: $u$ on a non-characteristic curve $M$ and $u_x$ along a non-characteristic curve $N$. We assume that $M$ and $N$ are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the $(x,y)$-plane. It is known that if $M$ is located above $N$, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when $M$ lies below $N$, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function $f$). The construction, via Picard iteration, makes use of a careful choice of additional $u$-data which are updated in each iteration step. This is a joint work with Kris Jenssen from Penn State University,

Title: A GBC Approache for Smooth Bijections between two domains.
Abstract: We present a construction of the bijection of the harmonic GBC map transforming from one arbitrary polygonal domain V to another arbitrary polygonal domain W. In addition, we shall point out that the harmonic GBC map is also a diffeomorphism over the interior of V to the interior of W. Finally, we remark on how to construct a harmonic GBC map from V to W when the number of vertices of V is different from the number of vertices of W and how to construct harmonic GBC functions over a polygonal domain with a hole or holes. We also point out that it is possible to use the harmonic GBC map to deform a nonconvex polygon V to another nonconvex polygon W by a good arrangement of the boundary map between ∂V and ∂W. Several numerical deformations based on images are presented to show the effectiveness of the map based on bivariate spline approximation of the harmonic GBC functions.

Title: Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies
Abstract: We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator \begin{equation} \label{eq_int_1} (\Delta^2-k^4 +q)u =0\quad \text{in}\quad \Omega \end{equation} from the knowledge of the partial Cauchy data set. Here $\Omega\subset \{x = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n: x_n<0\}$, $n\ge 3$, be a bounded open set with $C^\infty$ boundary. Assume that $\Gamma_0:=\partial \Omega\cap \{x_n=0\}$ is non-empty, and let us set $\Gamma=\partial \Omega\setminus \Gamma_0$. That is, our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established earlier, respectively. We establish H\”older type stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials, sharpening those of earlier results. In particular, no continuity of the potentials in the main result is assumed, and the required Sobolev regularity assumptions are fairly mild and are independent of the dimension.

Title: A la Dhont method for solving Stokes equation with prescribed boundary conditions
Abstract: The motion of a sphere in stratified fluids is of importance to environmental applications involving the ocean and the atmosphere. Modeling this problem involves solving the Navier-Stokes equation with variable density field. In the viscous limit, the Oseen tensor provides the fundamental function solution to the Stokes equations for the flow external to a sphere located at the origin. Motivated by the difficulty that arises from the three dimensional convolution of the Oseen tensor and the forcing term, we attempt to derive the solution through a different method that would be more suitable for numerical implementation. We computed a new solution for the fluid flow applying La Dhont Method to the Stokes equation with prescribed boundary conditions at the sphere and at infinity. Our new tensor is a result of solving the PDE and matching the restrictions of the boundary conditions, and the rate of convergence of the solution at infinity. Comparisons between the Oseen tensor and the new solution using La Dhont Method approach will be discussed.