In this dissertation, we will discuss some inverse problems associated to elliptic equations with rough coefficients.

The first results in this thesis are based on the papers of Haberman-Tataru [2013] and Haberman [2014]}, and relate to Calder on's problem for the conductivity equation

\[\div(\gamma \nabla u) = 0.\]

Given a domain $\Omega$ and a positive real-valued function $\gamma$ defined on $\Omega$, we may define the Dirichlet-to-Neumann map $\Lamba_\gamma$, which maps the Dirichlet data of solutions to the conductivity equation to the corresponding Neumann data. Calder on's problem is to determine $\gamma$ from $\Lambda_\gamma$. Uniqueness in Calder on's problem for smooth conductivities in dimensions three and higher was established by Sylvester and Uhlmann. Later, this result was extended by various authors to less regular conductivities which have at least one and a half derivatives. In Chapter 4, we will show that $\Lambda_\gamma$ uniquely determines a $C^1$ conductivity in dimensions three and higher. In Chapter 6, we will improve this result in dimensions $n=3,4$ to $\gamma \in W^{1,n}(\Omega)$. We also have a new result for conductivities in $W^{s,p}(\Omega)$ in higher dimensions for certain $(s,p)$ such that $s > 1$ but $W^{s,p}(\Omega) \not\subset W^{1,\infty}(\Omega)$.

In Chapter 7, we will prove a new result on recovering an unbounded magnetic potential $A$ from the Dirichlet-to-Neumann map $\Lambda_{A,q}$ for the Schr\"odinger equation

\[(D+A)^2 u+ qu = 0.\]

Previous results on this problem assumed that the potential $A$ is bounded. We will show that, in three dimensions, the integrability assumption on $A$ can be considerably relaxed in exchange for slightly more regularity. Namely, we will show that the Cauchy data uniquely determine $curl A$ if $q\in W^{-1,3}(\Omega)$ and $A$ is small in $W^{s,3}(\Omega)$, where $s>0$. We will also show that $q$ is determined by the Dirichlet-to-Neumann map if $A$ is small in $W^{s,3}(\Omega)$ and $q \in L^\infty(\Omega)$.

We consider the behavior of nonlinear KdV-type equations that admit quasilinear dynamics in the sense that the nonlinear flow cannot be simply treated as a perturbation of the linear flow, even for small initial data.

We treat two problems in particular. First we study the local dynamics of KdV-type equations with nonlinearities involving two spatial derivatives. A key obstruction to well-posedness arises from to the Mizohata condition. This leads to an additional integrability requirement for the solution in the absence of a suitable null structure. In this case we prove local well-posedness for large, low-regularity data in translation-invariant spaces.

Second we explore the global dynamics of the modified Korteweg de-Vries equation. We establish modified asymptotic behavior without relying on the integrable structure of the equation. This approach has the advantage that it can be used for a wide class of short-range perturbations of the mKdV. To give a thorough description of the asymptotic behavior we prove an asymptotic completeness result that relates mKdV solutions to the 1-parameter family of solutions to the Painlevé II equation.