Tbilisi Analysis & PDE Seminar

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Dear Colleagues! V. Kupradze Institute of Mathematics of the University of Georgia is pleased to invite you to the Online Tbilisi Analysis & PDE Seminar. The seminar is held bi-weekly on Wednesdays at 20:00 local time in Tbilisi (compare to your local time https://www.timeanddate.com/worldclock/georgia/tbilisi).

Talk on May 20, 2026

Speaker: Prof. Roland Duduchava,  Victor Kupradze Institute of Mathematics, The University of Georgia & A. Razmadze Mathematical Institute, Georgia;
E-mail:  roldud@gmail.com;
https://rmi.tsu.ge/~dudu/welcome_E.htm

The title of the lecture: "Convolution equations and BVPs for the Generic Laplacian on Lie groups"

Abstract: Lie group \( \mathbf{G} \) is a manifold where each element has the inverse \( x\circ x^{-1}=e \) and \( e \) is the neutral element (identity). Then on \( \mathbf{G} \) we have a unique invariant Haar measure \( d\mu_{\mathbf{G}} \), Fourier transform \( \mathcal{F}_{\mathbf{G}} \) with its inverse \( \mathcal{F}^{-1}_{\mathbf{G}} \), and associated generic differential operators \( \mathfrak{D}_1,\ldots,\mathfrak{D}_n \), generated by vector fields from the associated Lie algebra. We consider commutative (Abelian) Lie groups, homeomorphic to \( \mathbb{R}^n \) and, therefore, the dual group coincides with \( \mathbb{R}^n. \)

That allows definition of convolution integro-differential equations

\[
(W^0_{\mathbf{G},a}\varphi)(x)
=
(\mathcal{F}_{\mathbf{G}}^{-1}
\, a \,
\mathcal{F}_{\mathbf{G}}\varphi)(x)
=
\sum_{|\alpha|\le m}
\left[
c_{\alpha}\mathfrak{D}^{\alpha}\varphi(x)
+
\int_{\mathbf{G}}
k_{\alpha}(x\circ y^{-1})
\mathfrak{D}^{\alpha}\varphi(y)\, d_G y
\right]
=
f(x),
\quad x\in\mathbf{G}
\]

with the symbols of polynomial growth

and of Generic Bessel Potential Spaces \(\mathbb{G}\mathbb{H}^s_p(\mathbf{G},d_\mathbf{G} x)\).

In this framework we study convolution integro-differential equations in the setting

and Boundary Value Problems (BVPs) on domains \( \Omega\subset\mathbf{G} \) for the Generic Laplacian:

\[
\Delta_\mathbf{G}(\mathfrak{D})\psi(x)=h(x),
\quad
\Delta_\mathbf{G}:=\mathfrak{D}^2_1+\ldots+\mathfrak{D}^2_n
\quad
x\in\Omega,
\quad
\psi\in\mathbb{G}\mathbb{H}^s_p(\mathbf{G},d_\mathbf{G} x), \]

\[
(P(\mathfrak{D})\psi)^+(t)=g(t),
\quad
t\in\Gamma:=\partial\Omega,
\quad
h\in\mathbb{G}\mathbb{H}^{s-2}_p(\mathbf{G},d_\mathbf{G} x),\;\;
g\in\mathbb{G}\mathbb{H}^{s-r}_p(\mathbf{G},d_\mathbf{G} x)
\]

where \( P(\mathfrak{D}) \) is the operator of order \( r \), either the Dirichlet or Neumann trace operator on the boundary.

As an example we consider the Lie group \( I=(-1,1) \), where the group operation is \(x\circ y=(x+y)/(1+xy)\), the neutral element is \( 0 \), the inverse to \( x\in I \) is \( -x \), the Haar measure is \( d\mu_I=dx/(1-x^2) \), and the generic differential operator is \((1-x^2)\frac{d}{dx}\).

Download Poster: Link to Download

How to join: The seminar is organized on the Google Meet platform. If you are already registered, you do not need to register again. Otherwise, to join the seminar, please send an email to kim@ug.edu.ge or register here:

https://forms.gle/xfQJ9fg1uqe7CrZw6

You will then receive further information.

Google Meet Join Information

Organizers:
1. R. Duduchava, Institute of Mathematics, University of Georgia, Tbilisi, Georgia
2. E. Shargorodsky, Department of Mathematics, King’s College London, UK
3. A. Meskhi, Kutaisi International University, Kutaisi, Georgia

Secretary:
M. Tsaava, Institute of Mathematics, University of Georgia, Tbilisi, Georgia

Technical support:
G. Tutberidze, Institute of Mathematics, University of Georgia, Tbilisi, Georgia
T. Kuzmina, Institute of Mathematics, University of Georgia, Tbilisi, Georgia

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