170 research outputs found
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 , , is a
real-valued function and are real constants with
, is locally well-posed in for . In
the Hamiltonian case (\textit i.e. when ), the IVP associated to
\eqref{05KdV} is then globally well-posed in the energy space .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
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