Goldshtein, Vladimir

Pchelintsev, Evgeny A.

Ukhlov, Alexander

English

Tomsk State University Repository

Министерство науки и высшего образования РФИнститут математики им. С.Л. Соболева СО РАННовосибирский национальный исследовательскийгосударственный университетМеждународная конференция по геометрическомуанализу в честь 90-летия академика Ю.Г. Решетняка22–28 сентября 2019Тезисы докладовInternational Conference on Geometric Analysisin honor of the 90th anniversaryof academician Yu.G. Reshetnyak22–28 of September 2019AbstractsНовосибирск2019УДК 514+515.1+517ББК 22.16М431Международная конференция по геометрическому анализу в честь 90-летия академика Ю.Г. Решетняка, 22–28 сентября 2019: Тез. докл. / Подред. С. Г. Басалаева; Новосиб. гос. ун-т. — Новосибирск: ИПЦ НГУ, 2019. —164 с.International Conference on Geometric Analysis in honor of 90th anniversaryof academician Yu.G. Reshetnyak, 22–28 of September 2019: Abstracts / ed. byS. G. Basalaev; Novosibirsk State University. — Novosibirsk: PPC NSU, 2019. —164 p.ISBN 978-5-4437-0949-9Сборник содержит тезисы некоторых докладов, представленных на Меж-дународной конференции по геометрическому анализу, проводимой в честь90-летия академика Ю.Г. Решетняка (22–28 сентября 2019 года). Темы до-кладов относятся к современным направлениям в геометрии, теории управ-ления и анализе, а также к приложениям методов метрической геометрии ианализа к смежным областям математики и прикладным задачам.Мероприятие проведено при финансовой поддержке Российского фондафундаментальных исследований (проект № 19-01-20122) и Регионального ма-тематического центра НГУ.The digest contains abstracts of some of the talks presented on the Inter-national Conference on Geometric Analysis in honor of the 90th anniversaryof academician Yu. G. Reshetnyak (22–28 of September, 2019). Topics of talksconcern modern trends in geometry, control theory and analysis, as well as ap-plications of the methods of the metric geometry and analysis to related fields ofmathematics and applied problems.The conference is supported by Russian Foundation for Basic Research (projectN 19-01-20122) and NSU Regional Mathematical Center.УДК 514+515.1+517ББК 22.16c○ Новосибирский государственныйISBN 978-5-4437-0949-9 университет, 2019The mixed volumes of convex hull of distinct leaves can be obtained by thepolarization of the polynomial 𝑉 (𝑝). According to SP, the coefficient at 𝑠𝑘𝑡8−𝑘is equal to the multiplied by(︀8𝑘)︀mixed volume of 𝑘 bodies ̂︀𝐹𝜛1 and 8− 𝑘 bodieŝ︀𝐹𝜛2 , where 𝜛1, 𝜛2 are the fundamental weights (see [4, 5.1.26]).Due to BKK-theory we know the explicit formula for Be´zout’s number (i.e.the number of solutions) for systems of equations 𝑓𝑘 = 0 with matrix elements offinite dimensional representations of a complex reductive Lie group. It involvesvolumes and mixed volumes of the convex hulls of integral orbits of the coadjointrepresentations (see [5]). We are grateful to Boris Kazarnovskiˇı for making usaware of this fact and for useful comments. These results were extended andgeneralized in several directions but to the best of our knowledge the case ofisoparametric foliations is not covered yet.References1. Gichev V.M. Polar representations of compact groups and convex hulls of theirorbits, Diff. Geom. Appl. 28 (2010), 608–614.2. Palais R. S., Terng C.-I. Critical Point Theory and Submanifold Geometry, Springer-Verlag Berlin Heidelberg, 1988.3. Thorbergsson G. Isoparametric foliations and their buildings, Annals of Math.133:2 (1991), 429–446.4. Schneider R. Convex bodies, Brunn-Minkowski theory, Cambridge, 1993.5. Казарновский Б.Я. Многогранники Ньютона и формула Безу для матрич-ных функций конечномерных представлений, Функц. анализ и его прил. 21:4(1987), 73–74.Composition operators on Sobolev spaces andthe spectrum of elliptic operatorsVladimir Gol’dshtein𝑎, Valerii Pchelintsev𝑏,* Alexander Ukhlov𝑎𝑎Ben-Gurion University of the Negev;𝑏Tomsk Polytechnic University, Tomsk State Universityemail: vladimir@math.bgu.ac.il, vpchelintsev@vtomske.ru,ukhlov@math.bgu.ac.ilThis work is devoted to applications of the geometric theory of compositionoperators on Sobolev spaces to spectral problems of the 𝐴-divergent form ellipticoperators with the Neumann boundary condition:𝐿𝐴 = −div[𝐴(𝑧)∇𝑓(𝑧)], 𝑧 = (𝑥, 𝑦) ∈ Ω, 𝜕𝑓𝜕𝑛⃒⃒⃒⃒𝜕Ω= 0, (1)*Research is supported by RFBR Grand No. 18-31-00011.56in quasiconformal regular domains Ω ⊂ C with 𝐴 ∈ 𝑀2×2(Ω). We denote, by𝑀2×2(Ω), the class of all 2 × 2 symmetric matrix functions 𝐴(𝑧) = {𝑎𝑘𝑙(𝑧)},det𝐴 = 1, with measurable entries satisfying the uniform ellipticity condition1𝐾|𝜉|2 ≤ ⟨𝐴(𝑧)𝜉, 𝜉⟩ ≤ 𝐾|𝜉|2 a.e. in Ω, (2)for every 𝜉 ∈ C, where 1 ≤ 𝐾 <∞.Recall that a simply connected domain Ω ⊂ C is called an 𝐴-quasiconformal𝛽-regular domain, 𝛽 > 1, if∫︁∫︁D|𝐽(𝑤,𝜙−1)|𝛽 𝑑𝑢𝑑𝑣 <∞,where 𝜙 : Ω → D is an 𝐴-quasiconformal mapping [3]. Since (see, for exam-ple, [1]) 𝐴-quasiconformal mappings 𝜙 : Ω → D are defined up to conformalautomorphisms of D, this definition doesn’t depend on a choice of 𝜙 and dependson the quasihyperbolic geometry of Ω only. A class of quasiconformal regulardomains includes Lipschitz domains, Gehring domains and also some fractal do-mains like snowflakes.The suggested method is based on the connection between composition oper-ators on Sobolev spaces and quasiconformal mappings, that refines (in the case𝑛 = 2) results of [2]. As applications we obtain the solvability of the spectralproblem (1) and lower estimates of the first non-trivial Neumann eigenvalues in𝐴-quasiconformal 𝛽-regular domains [3]:Theorem. Let 𝐴 be a matrix satisfies the uniform ellipticity condition (2) andΩ be an 𝐴-quasiconformal 𝛽-regular domain. Then the spectrum of the Neumanndivergence form elliptic operator 𝐿𝐴 in Ω is discrete, and can be written in theform of a non-decreasing sequence:0 = 𝜇0(𝐴,Ω) < 𝜇1(𝐴,Ω) ≤ 𝜇2(𝐴,Ω) ≤ . . . ≤ 𝜇𝑛(𝐴,Ω) ≤ . . . ,and1𝜇1(𝐴,Ω)≤ 4𝛽√𝜋(︂2𝛽 − 1𝛽 − 1)︂ 2𝛽−1𝛽(︂∫︁∫︁D|𝐽(𝑤,𝜙−1)|𝛽 𝑑𝑢𝑑𝑣)︂ 1𝛽,where 𝐽(𝑤,𝜙−1) is a Jacobian of the quasiconformal mapping 𝜙−1 : D→ Ω.References1. Astala K., Iwaniec T., Martin G. Elliptic partial differential equations and quasi-conformal mappings in the plane, Princeton University Press (2008).2. Vodop’yanov S.K., Gol’dstein V.M. Lattice isomorphisms of the spaces 𝑊 1𝑛 andquasiconformal mappings, Siberian Math. J. 16:2 (1975), 174–189.3. Gol’dshtein V., Pchelintsev V., Ukhlov A. Composition operators on Sobolevspaces and eigenvalues of divergent elliptic operators, arXiv:1903.11301.57

Composition operators on Sobolev spaces and the spectrum of elliptic operators

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Composition operators on Sobolev spaces and the spectrum of elliptic operators

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