The three short courses consist of three 90 minutes lectures.
Preliminary List of Confirmed Invited Speakers
Preliminary List of Confirmed Invited Speakers
Martin Costabel, Rennes, France
Area: Classes of Pseudodifferential Operators
Title:The Cosserat eigenvalue problem
The Cosserat eigenvalue problem is the Dirichlet problem for the Lamé equations of linear elasticity, where the Lamé parameter lambda (bulk modulus) is considered as the eigenvalue parameter. Estimates for the Cosserat eigenvalues are related to the Korn inequality and to the inf-sup constant for the divergence. The problem has a long history, starting with E. and F. Cosserat (1898), and contributions by Friedrichs (1937) and Mikhlin (1973). It has recently got more attention mainly from people in fluid dynamics, but also in electrodynamics and finite element analysis. Although there has been some progress, in particular for domains with corners, many very simple questions on this problem are still open, like the precise value of the lowest Cosserat eigenvalue for a square or a triangle.
In the talks, I will present the problem and its relations to some other problems of vector analysis and describe the classsical results for smooth domains. Then I will show how Mellin analysis gives the essential spectrum for corner domains and present some computations for rectangles. Finally a relation with the spectral theory for a strongly singular volume integral operator in electromagnetism is analyzed.
Title:The Cosserat eigenvalue problem
The Cosserat eigenvalue problem is the Dirichlet problem for the Lamé equations of linear elasticity, where the Lamé parameter lambda (bulk modulus) is considered as the eigenvalue parameter. Estimates for the Cosserat eigenvalues are related to the Korn inequality and to the inf-sup constant for the divergence. The problem has a long history, starting with E. and F. Cosserat (1898), and contributions by Friedrichs (1937) and Mikhlin (1973). It has recently got more attention mainly from people in fluid dynamics, but also in electrodynamics and finite element analysis. Although there has been some progress, in particular for domains with corners, many very simple questions on this problem are still open, like the precise value of the lowest Cosserat eigenvalue for a square or a triangle.
In the talks, I will present the problem and its relations to some other problems of vector analysis and describe the classsical results for smooth domains. Then I will show how Mellin analysis gives the essential spectrum for corner domains and present some computations for rectangles. Finally a relation with the spectral theory for a strongly singular volume integral operator in electromagnetism is analyzed.
Roland Duduchava, Tbilissi, Georgia
Area: Boundary Value Problems
Title: Calculus of partial differential operators on hypersurfaces and shell theory
Partial differential equations on Riemannian manifolds are usually written in intrinsic coordinates, involving metric tensor and Christoffel symbols. But we deal with a hypersurface, the Cartesian coordinates from the ambient space can be applied. The calculus is based on the Gunters derivatives, which are the projections of the classical coordinate derivatives from the ambient Euclidean space.
Our scope in the present series of lectures is to deliver the recent results on the shell theory. It is well known that the theory of thin linearly elastic shells leads to two-dimensional equations on the middle hypersurface of the shell. We should apply the developed calculus of Gunter's derivatives and prove that when the thickness of the shell converges to 0, the equation of 3D elasticity converges exactly to the Lamé equation with constant Lamé coefficients on the middle surface. It is remarkable, that the applied asymptotic analysis allows to derive high order limit equations on the middle surface, which have the same simple form.
The limit equation obtained by this process is much simpler than the one derived by Koiter-Sanchez-Palenzia-Ciarlet etc. with a similar asymptotic analysis, but based on the classical differential geometry.
Title: Calculus of partial differential operators on hypersurfaces and shell theory
Partial differential equations on Riemannian manifolds are usually written in intrinsic coordinates, involving metric tensor and Christoffel symbols. But we deal with a hypersurface, the Cartesian coordinates from the ambient space can be applied. The calculus is based on the Gunters derivatives, which are the projections of the classical coordinate derivatives from the ambient Euclidean space.
Our scope in the present series of lectures is to deliver the recent results on the shell theory. It is well known that the theory of thin linearly elastic shells leads to two-dimensional equations on the middle hypersurface of the shell. We should apply the developed calculus of Gunter's derivatives and prove that when the thickness of the shell converges to 0, the equation of 3D elasticity converges exactly to the Lamé equation with constant Lamé coefficients on the middle surface. It is remarkable, that the applied asymptotic analysis allows to derive high order limit equations on the middle surface, which have the same simple form.
The limit equation obtained by this process is much simpler than the one derived by Koiter-Sanchez-Palenzia-Ciarlet etc. with a similar asymptotic analysis, but based on the classical differential geometry.
Ilya Spitkovsky, Williamsburg, Virginia USA
Area: Factorization Theory
Title: Advances in factorization theory
Factorization of matrix functions is a rich and still evolving theory, with applications to mathematical physics, integrable systems, random matrices, sampling and interpolation, to name a few. Two main problems in this theory are: (i) factorability criteria for matrix functions from various particular classes, and (ii) constructive and explicit factorization algorithms and formulas. In this series of lectures, we will attempt to systematize and report the developments in these two areas, with the emphasis on the progress in the last ten years. In particular, we will address the factorization problem in the ordered abelian group setting, allowing in particular to consider the classical Wiener-Hopf factorization and the almost periodic factorization from the unified point of view.
Title: Advances in factorization theory
Factorization of matrix functions is a rich and still evolving theory, with applications to mathematical physics, integrable systems, random matrices, sampling and interpolation, to name a few. Two main problems in this theory are: (i) factorability criteria for matrix functions from various particular classes, and (ii) constructive and explicit factorization algorithms and formulas. In this series of lectures, we will attempt to systematize and report the developments in these two areas, with the emphasis on the progress in the last ten years. In particular, we will address the factorization problem in the ordered abelian group setting, allowing in particular to consider the classical Wiener-Hopf factorization and the almost periodic factorization from the unified point of view.
