Analysis Seminar

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: In this talk I will discuss recent results regarding regularity properties
of Green functions associated with elliptic differential operators of second and higher
order in irregular domains. This analysis includes the case of second and higher
order elliptic systems with constant coefficients, the bi-Laplacian, and the Stokes
system.

Abstract: I will describe results obtained in collaboration with H. Jacobowitz
some years ago and recently with A. Bergamasco and S. Zani on topological con-
sequences of assumptions made on differential operators. This is a problem of a
global nature: the objects on which the operators act are functions (more generally
sections of a complex vector bundle) on a compact manifold without boundary.
The analytic conditions are either ellipticity, or global hypoellipticity. I will also
pose an open problem concerning solvability in the large.

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.

Abstract: In our work we present a modification of the known Baouendi-Treves
Approximation Theorem. Instead of working with a general $N$ -dimensional smooth
manifold we will use a quadric manifold $M$.
While the original theorem deals with any locally integrable structure $\mathcal{L}$ of $\mathbb{C}T M$
we will focus on the CR-structure of $M$ and its solutions: the CR distributions.
With this restrictions we achieve convergence in any compact set instead of local
approximation. More precisely: our main theorem states that if $u\in$CR$(M)$ then one can find
smooth CR-polynomials $P_{n}(w, t)$ such that for every compact subset $K$ we can
approximate $u$ by $P_{n}$ in the distribution sense. In addition to that
(1) if $u\in C^{k}(M )$, $k = 0, 1,\ldots$, then the convergence occurs in the topology of $C^{k}(K)$;
(2) if $u\in h^{p}(M )$, $0 < p < \infty$, then the approximation occurs in the topology of $h^{p}(K)$.
If there is enough time, we will present an application.

Abstract: I will give an introduction to the renormalization theory of area-preserving maps, and present several recent results about the dynamics in a neighbourhood of the fixed point of the period doubling renormalization operator: $F\to\Lambda^{−1}\circ F\circ F\circ\Lambda$, where $\Lambda$ is a diagonal matrix. The results are based on a detailed study of the spectral properties of the derivative of the renormalization operator. The talk is based on joint work with Denis Gaidashev and Marco Martens.

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: After an introduction concerning the fractional Laplace operators and
the k-th Hessian operators, some relations between the k-Hessian energy and the
fractional Sobolev energy will be presented.

Our starting point is a celebrated approximation theorem due to M.
Artin (1969) which roughly states that any formal solution of a system
of polynomial equations can be approximated (in the Krull topology)
by a sequence of algebraic solutions. Our goal in this talk is to explore
whether a similar conclusion holds when the system of polynomial is
coupled with certain (linear) pde’s. We will discuss the class of linear operators for which such a generalization is possible. These opera-
tors arise as tangential Cauchy-Riemann operators of real-algebraic CR
manifolds (and include, as a very special case, the standard Cauchy-
Riemann operator of the complex euclidean space).

Abstract: The plan of the talk is to describe some interesting trigonometric series
which we call “iterated Fourier series” and their connections to natural problems
of “physical reality”.

Abstract: We study the asymptotic behavior at infinity of solutions of a perturbed Dirac Operator and their relation to the outgoing and incoming solutions of
the Helmholtz equation and the Maxwell system in terms of their far-field patterns
(joint work with Salvador Perez-Esteva).

Abstract: The transmission eigenvalue problem is a new class of eigenvalue
problems that has recently appeared in inverse scattering theory for inhomogeneous media. Such eigenvalues provide information about material properties of
the scattering object and can be determined from scattering data, hence can play
an important role in a variety of problems in target identification. The transmission
eigenvalue problem is non-selfadjoint and nonlinear which make its mathematical
investigation very interesting.
In this lecture we will describe how the transmission eigenvalue problem arises
in scattering theory, how transmission eigenvalues can be computed from scattering
data and what is known mathematically about these eigenvalues. The investigation of transmission eigenvalue problem for anisotropic media will be discussed
and Faber-Krahn type inequalities for the first real transmission eigenvalue will be
presented.
We conclude our presentation with some recent preliminary results on transmission eigenvalues for absorbing and dispersive media, i.e. with complex valued index
of refraction, as well as for anisotropic media with contrast that changes sign.
Our presentation contains a collection of results obtained with several collaborators, in particular with David Colton, Drossos Gintides, Houssem Haddar and
Andreas Kirsch.

Abstract: A new proof of both analytic and $C^{\infty}$− hypoellipticity of Kohn’s
operator is given using FBI techniques introduced by J. Sjostrand. The same proof
allows us to obtain both kinds of hypoellipticity at the same time. We prove also
the hypoellipticity in the sense of germs, even though it fails to be hypoelliptic
in the strong sense, for an operator obtained by “slightly perturbing” the Kohn’s
operator.