Recent progress in the Calderon problem with partial data

We survey recent results on Calderon's inverse problem with partial data, focusing on three and higher dimensions.Comment: 36 page

Free boundary regularity for harmonic measures and Poisson kernels

One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on \R^{n+1} is close to the doubling constant of the n-dimensional Lebesgue measure then its support is well approximated by n-dimensional affine spaces, provided that the support is relatively flat to start with. Primarily we consider sets which are boundaries of domains in \R^{n+1}. The n-dimensional Hausdorff measure may not be defined on the boundary of a domain in R^{n+1}. Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions. We obtain a new characterization of locally flat domains in terms of the doubling properties of their harmonic measure. Along these lines we investigate how the ``weak'' regularity of the Poisson kernel of a domain determines the geometry of its boundary.Comment: 85 pages, published version, abstract added in migratio

Well-posedness for the fifth-order KdV equation in the energy space

We prove that the initial value problem (IVP) associated to the fifth order KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3), {equation} where $x \in \mathbb R$, $t \in \mathbb R$, $u=u(x,t)$ is a real-valued function and $\alpha, \ c_1, \ c_2, \ c_3$ are real constants with $\alpha \neq 0$, is locally well-posed in $H^s(\mathbb R)$ for $s \ge 2$. In the Hamiltonian case (\textit i.e. when $c_1=c_2$), the IVP associated to \eqref{05KdV} is then globally well-posed in the energy space $H^2(\mathbb R)$.Comment: We corrected a few typos and fixed a technical mistake in the proof of Lemma 6.3. We also changed a comment on the work of Guo, Kwak and Kwon on the same subject according to the new version they posted recently on the arXiv (arXiv:1205.0850v2