We study the problem of noisy data fitting and spectral measure recovery in the classes Stieltjes and Completely monotone functions. The analytical setting is least squares over the convex or conical hull of a space curve in IR^n. I will begin with the case of least squares over convex hulls where the convergence analysis is simple. Here, I will introduce our method of analyzing the support of the optimal spectral measure. I will then move to the case of conical hulls (non-negative least squares). In this context, convergence analysis is not well understood. We introduce a simple assumption on the problem which allows us to overcome the convergence problem. While our assumption does not initially hold in either our motivating examples, I will show they can be rescaled so that is does hold. This rescaling allows certain algorithms for convex minimization to be extended to minimization problems with non-negativity constraints. I will provide several numerical examples which show convergence of algorithms as well as highlight our spectral measure analysis at certain noise levels.

Abstract: I will consider the focusing NLS equation in one, two and three space dimensions with different powers of nonlinearities (including cubic and quintic powers) and their global solutions with finite energy $H^{1}$ initial data. My discussion will focus on blow up solutions and known types of their dynamics. In particular, I will show that the class of so-called ‘log-log’ blow up solutions can blow up not only on a single point set but on various geometric sets such as circles, spheres, cylinders, while remaining regular (in the energy space) away from the blow-up core.

Abstract: In 1979 Joseph J. Kohn defined ideal sheaves of multipliers and an algorithm for producing these in order to investigate the subellipticity of the $\overline{\partial}$ Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will be discussing the properties of these sheaves in the cases when the boundary is smooth, real-analytic, and Denjoy-Carleman. I will show that in the smooth case these ideal sheaves are quasi-flasque, and I will discuss coherence in the real-analytic case. The Denjoy-Carleman case is intermediate between the two, and I will show to what extent the nice properties of the real-analytic case transfer over.

Abstract: We study static 180 degree domain walls in infinite thin magnetic wires with either a rectangular or a centrally symmetric Lipschitz cross section. In dependence of the thickness of the wire, different pattern formations of the magnetization vector are observed. We prove an existence of global minimizers (even for Lipschitz cross sections). We prove a $\Gamma$-convergence result for both types of thin wires. For rectangular cross sections we distinguish two different regimes and establish the minimal energy scaling in terms of the cross section edge’s lengths. For a centrally symmetric cross section we establish as well the minimal energy scaling in terms of the diameter of the cross section and some geometric parameters relating to it. We prove as well a rate of convergence for the minimal energies for all cases and an $H^{1}$ convergence for almost minimizers. For thick wires with a rectangular cross section we prove an upper bound and give a reference for a lower bound on the minimal energy. For thin wires a Nel wall occurs and for thick wires a vortex wall is expected to occur.

Abstract: In joint work with Federico Tournier, we obtain an invariant Harnacks inequality for non negative solutions to degenerate elliptic equations of the form \[a(x; y; z)X1; 1u + 2b(x; y; z)X1; 2u + c(x; y; z)X2; 2u = 0,\] where $Xi; j$ are dened with the Heisenberg vector fields and the matrix coefficient is uniformly elliptic, and satisfying the additional condition that the ratio between the maximum and minimum eigenvalues is sufficiently close to one. In the paper we prove critical density and double ball estimates, once this is established, Harnack follows directly from the results of Gutierrez, Lanconelli and Di Fazio, Mathema- tishe Annalen, 2008. Preprint available at http://math.temple.edu/~gutierre/papers/harnack.subelliptic.final.version.june.28.2011.pdf

Abstract: The Levi curvatures for a real hypersurface of $\mathbb{C}^{n+1}$ can be defined in analogy with Euclidean curvatures. The operator related to these curvatures is a second order fully non-linear operator. Its characteristic form, when computed on some ”pseudoconvex” function, is non-negative definite with kernel of dimension one. Since the missing ellipticity direction can be recovered by a suitable commutation relation, a strong comparison principle holds. This is an important tool for identification results. Using a technique introduced by Hounie and Lanconelli, we study bounded Reinhardt domains of $\mathbb{C}^{2}$ with an assigned curvature reflecting its symmetries.

Abstract: We define the notion of regularity of traces of divergence-free fields. We then show that such regularity is a property of all strong local minimizers. Next we prove that the regularity of traces coupled with classical necessary conditions for strong local minima is sufficient for attainment of the minimum.