LiU / Matematiska Institutionen

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Ett informationsblad från matematiska institutionen vid Linköpings universitet. Material till Lite Mat lämnas till Bodil Stavklint litemat@mai.liu.se senast torsdagar kl 08.00.

Vi har dessutom ett arkiv av gamla nummer.

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***LITE MAT***

Ett informationsblad från matematiska institutionen vid Linköpings universitet

v39 2004

Matematiska kollokviet

Onsdagen den 22 september, kl. 13.00-14.00 talar Professor Henrik Shahgholian, KTH,

The structure of the singular set of a free boundary in potential theory.

Sammanfattning: We characterize the structure of the singular set in the following free boundary problem

$\displaystyle (\Delta u -f )u =0, \qquad \qquad \hbox{in } \quad B=B(0,1),$

where $ f$ is Lipschitz, and $ u\in W^{2,p}(B)$, $ p>n$. The free boundary $ \partial \Omega$, represented by $ \partial \{ \Delta u =f\}$, appears in certain problems in geophysics and inverse problems in potential theory.

This is joint work with Luis Caffarelli.

Onsdagen den 6 oktober, kl. 13.00-14.00 talar Professor Natan Krugljak, Luleå,

On one new covering theorem and its applications.

Sammanfattning: Classical covering theorems (Vitali, Whitney, Besicovitch) are not only beautiful but they also have important applications in analysis, harmonic analysis, theory of approximations, ergodic theory and PDEs. Some years ago in connection with real interpolation of Sobolev spaces appeared theorems which have simultaniously features of Whitney and Besicovitch covering theorems. I plan to discuss these new theorems and their applications to interpolation and possible applications to singular integrals.

Lokal: ISY/MAI:s seminarierum Glashuset, ing 25.

Välkomna!
Anders Björn,
Vladimir Kozlov,
Svante Linusson och
Stefan Rauch-Wojciechowski

Licentiatseminarier

Onsdagen den 22 september kl 10.15 försvarar Peter Rand sin licentiatavhandling med titeln

Asymptotic analysis of a nonlinear partial differential equation in a semicylinder.

Opponent är professor Henrik Shahgholian från KTH.

Abstract: We study small solutions of a nonlinear partial differential equation in a semi-infinite cylinder. The asymptotic behaviour of these solutions at infinity is determined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vanishes at infinity or tends to a nonzero periodic solution of a nonlinear ordinary differential equation. Thereafter, the same equation under the Dirichlet boundary condition is studied, but now the nonlinear term and right-hand side are slightly more general than in the Neumann problem. Here, an estimate of the solution in terms of the right-hand side of the equation is given. If the equation is homogeneous, then every solution small enough tends to zero. Moreover, if the cross-section is star-shaped and the nonlinear term in the equation is subject to some additional constraints, then every bounded solution of the homogeneous Dirichlet problem vanishes at infinity. An estimate for the solution is given.

Lokal: Glashuset.

Välkomna!

Fredagen den 24 september kl 10.15 försvarar Jonas Bergman sin licentiatavhandling med titeln

Conformal Einstein spaces and Bach tensor generalizations in $ n$ dimensions.

Platsen är Glashuset och opponent är Docent Mattias Marklund, Institutionen för fysik, Umeå Universitet.

Abstract: In this thesis we investigate necessary and sufficient conditions for an $ n$-dimensional space, $ n \geq 4$, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an $ n$-dimensional generalization of the Bach tensor $ B{}_{ab}$ with the same properties. We strengthen a theorem due to Belfagón and Jaén and give a basis ($ U{}_{ab}$, $ V{}_{ab}$ and $ W{}_{ab}$) for all $ n$-dimensional symmetric, divergence-free 2-index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship $ B{}_{ab} = \frac{1}{2} U{}_{ab} + \frac{1}{6}V{}_{ab}$ and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.

Nevertheless, we are able to generalize the four-dimensional results due to Kozameh, Newman and Tod to $ n$ dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities (some of which are newly derived in this thesis) is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities.

For $ n$ dimensions, we define the tensors $ \mathfrak{b}_{abc}$ and $ \mathfrak{B}_{ab}$, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space.

Välkomna!

Seminarium i beräkningsvetenskap

Torsdagen den 23 september kl 10.15 talar I. A. Abrikosov, professor i teoretisk fysik, IFM, över ämnet

Quantitative quantum description of materials properties: Physics and Mathematics.

Abstract: Quantum mechanics revolutionized physics in the beginning of the last century. In 1927 Sommerfeld applied the quantum description to metals, within the model of free electrons, and obtained a qualitatively correct picture of some basic properties. But due to an enormous complexity associated with quantum mechanical calculations for real materials accurate quantitative results were rarely obtained. The possibilities to study materials properties from the basic principles of quantum mechanics were enormously enhanced when the density functional theory (DFT) and the local spin density approximation (LSDA) were formulated by Kohn and co-workers in the mid-60's. In 1998 this groundbreaking theory was awarded the Nobel Prize. I will give very brief and informal description of the DFT equations. I will also explain general ideas on how one solve numerically the equations within the DFT. I will start with variational principle, and show how one derives the so-called secular equation, the main equation to be solved numerically. I will particularly point out to unsolved mathematical/numerical problems that we are dealing with in our research.

Lokal: Kompakta rummet

Välkomna
Lars Eldén

Jämställdhetsseminarium

Onsdag 13 oktober kl 15-17 arrangerar MAI:s jämställdhetsgrupp ett seminarium med Christer Knuthammar, ordförande i universitetets jämställdhetsråd, på temat

"LiU och jämställdhet - visioner och verklighet."

Christer Knuthammar bidrar med reflektioner som grund för diskussion och eftertanke. Därefter följer gemensam fika och diskussion. Ingen föranmälan krävs.

Plats: Glashuset.

Varmt välkommen!
MAI:s jämställdhetsgrupp

Arbetsmiljöombud

Nytt arbetsmiljöombud för MAI är Theresia Roth, throt@mai.liu.se.



Mer information om MAI finns på MAIs hemsida.
Material till Lite Mat lämnas till Bodil Stavklint senast torsdagar kl 08.00.
Linköpings universitet, 581 83 Linköping
Tel 013-281000, Fax 013-100746
E-mail: litemat@mai.liu.se


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Sidan underhålls av: LiteMat litemat@mai.liu.se
Senast ändrad: Fri 2004-09-17; 17:53 MEST