# A scientific tribute to Prof. Nikolai Tarkhanov

## 23.03.1956-31.08.2020

### September 18th, 2020 (online meeting)

### Abstracts

- Essmaail Alsbikhan (Univ. of Potsdam)

*The Rouché theorem for operator functions*
**Abstract:** We derive a generalization of Rouché's theorem for operator functions. This talk is based on an ongoing Master thesis under the former supervision of Prof. Tarkhanov.

- Oleg Kiselev (Russian Academy of Sciences, Ufa)

*Captured particles in resonance at a potential well with a dissipation*
**Abstract:** We study the structure of the manifold in the phase space of captured particles in
resonance at a potential well with a dissipative perturbation and external periodic excitation.
The measure in the phase space of resonance solutions is studied.

- Ihsane Malass (Univ. of Potsdam)

*A perturbation of the De Rham complex*
**Abstract:** We perturb the de Rham complex by a non-closed 1-form "a" yielding a quasi
complex with differential ( d+a). In this context we investigate classical tools borrowed from
Lefschetz theory, Hodge Theory, as well as the analytic torsion. This presentation is based
on an article with the same title co-authored with Pr. Tarkhanov to appear in The Journal
of Siberian Federal University: Mathematics and Physics.

- Bert-Wolfgang Schulze (Univ. of Potsdam)

*Analysis on Manifolds with Singularities*
**Abstract:** We consider elliptic operators on stratified spaces *M* ∈ 𝔐_{k} with regular singularities of higher order *k*. The case of boundary value problems (BVPs) corresponds to order *k* = 1. Boutet de Monvel's calculus of BVPs with the transmission property is a special case. The operators with violated transmission property are covered by the edge calculus (*k* = 1). Singularities of higher order *k* give rise to a sequence of strata *s*_{j}(M), j = 0,..,k. Operators *A* in corresponding algebras of operators (corner-degenerate in stretched variables) are determined by a hierarchy of symbols *σ*_{j}(A), j = 0,..,k, modulo lower order terms. Those express ellipticity and parametrices *A*^{(-1)} in weighted corner Sobolev spaces, containing sequences of real weights *γ*_{j}. Higher symbol components depending on variables in *T*^{*}(s_{j}(M))\0 for *j > 0* act as operator families on infinite straight cones with singular links, and *σ*_{0}(A) is the standard principal symbol on *T*^{*}(s_{0}(M))\0. Clearly there is also the concept of elliptic complexes on *M*.

- Ivan Shestakov (Univ. of Oldenburg)

*Asymptotics of solutions to the Laplace-Beltrami equation on a rotation surface with a cusp*
**Abstract:** We discuss an asymptotic behaviour of solutions to the Laplace-Beltrami operator on a rotation surface near a cuspidal point. To this end we use the WKB-approximation.
This approach describes the asymptotic behaviour of the solution more explicitly than ab-
stract theory for operators with operator-valued coefficients. This is joint work with Prof. O.
Kiselev.

- Alexander Shlapunov (Siberian Federal Univ.)

*Existence theorems for regular spatially periodic solutions to the Navier-Stokes equations*
**Abstract:** We consider the initial problem for the Navier-Stokes equations over R3 x [0,T]
with a positive time T in the spatially periodic setting. Identifying periodic vector-valued
functions on R^{3} with functions on the 3-dimensional torus T^{3}, we prove that the problem
induces an open both injective and surjective mapping of specially constructed function spaces
of Bochner-Sobolev type.This gives a uniqueness and existence theorem for regular solutions
to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image
by estimating all possible divergent sequences in the preimage and matching the asymptotics.

- Sergey Vodopyanov (Novosibirsk State University)

*Mappings with bounded (q, p)-distortion and non-linear elasticity theory*
**Abstract:** The talk is devoted to interconnection of two subjects: composition operators Sobolev spaces, and non-linear elasticity theory.

The talk is based on papers in

1. Vodopyanov, S.K., Ukhlov, A.D. : Sobolev spaces and (P, Q)-quasiconformal mappings of Carnot groups. Sib. Math. J. 39(4), 665–682 (1998)

2. Molchanova A., Vodopyanov S. Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity. Calc. Var. (2020) 59:17 https://doi.org/10.1007/s00526-019-1671-4