Eurasian mathematical journal

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ISSN 2077–9879

Eurasian

Mathematical Journal

2016, Volume 7, Number 3

Founded in 2010 by

the L.N. Gumilyov Eurasian National University in cooperation with

the M.V. Lomonosov Moscow State University the Peoples’ Friendship University of Russia

the University of Padua

Supported by the ISAAC

(International Society for Analysis, its Applications and Computation) and

by the Kazakhstan Mathematical Society

Published by

the L.N. Gumilyov Eurasian National University

Astana, Kazakhstan

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Editorial Board

Editors–in–Chief

V.I. Burenkov, M. Otelbaev, V.A. Sadovnichy

Editors

Sh.A. Alimov (Uzbekistan), H. Begehr (Germany), T. Bekjan (China), O.V. Besov (Rus- sia), N.A. Bokayev (Kazakhstan), A.A. Borubaev (Kyrgyzstan), G. Bourdaud (France), A. Caetano (Portugal), M. Carro (Spain), A.D.R. Choudary (Pakistan), V.N. Chubarikov (Russia), A.S. Dzumadildaev (Kazakhstan), V.M. Filippov (Russia), H. Ghazaryan (Arme- nia), M.L. Goldman (Russia), V. Goldshtein (Israel), V. Guliyev (Azerbaijan), D.D. Haroske (Germany), A. Hasanoglu (Turkey), M. Huxley (Great Britain), M. Imanaliev (Kyrgyzstan), P. Jain (India), T.Sh. Kalmenov (Kazakhstan), B.E. Kangyzhin (Kazakhstan), K.K. Ken- zhibaev (Kazakhstan), S.N. Kharin (Kazakhstan), E. Kissin (Great Britain), V. Koki- lashvili (Georgia), V.I. Korzyuk (Belarus), A. Kufner (Czech Republic), L.K. Kussainova (Kazakhstan), P.D. Lamberti (Italy), M. Lanza de Cristoforis (Italy), V.G. Maz’ya (Swe- den), E.D. Nursultanov (Kazakhstan), R. Oinarov (Kazakhstan), K.N. Ospanov (Kaza- khstan), I.N. Parasidis (Greece), J. Peˇcari´c (Croatia), S.A. Plaksa (Ukraine), L.-E. Pers- son (Sweden), E.L. Presman (Russia), M.A. Ragusa (Italy), M.D. Ramazanov (Russia), M. Reissig (Germany), M. Ruzhansky (Great Britain), S. Sagitov (Sweden), T.O. Sha- poshnikova (Sweden), A.A. Shkalikov (Russia), V.A. Skvortsov (Poland), G. Sinnamon (Canada), E.S. Smailov (Kazakhstan), V.D. Stepanov (Russia), Ya.T. Sultanaev (Russia), I.A. Taimanov (Russia), T.V. Tararykova (Great Britain), J.A. Tussupov (Kazakhstan), U.U. Umirbaev (Kazakhstan), Z.D. Usmanov (Tajikistan), N. Vasilevski (Mexico), Dachun Yang (China), B.T. Zhumagulov (Kazakhstan)

Managing Editor

A.M. Temirkhanova

c

The Eurasian National University

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Aims and Scope

The Eurasian Mathematical Journal (EMJ) publishes carefully selected original research papers in all areas of mathematics written by mathematicians, principally from Europe and Asia. However papers by mathematicians from other continents are also welcome.

From time to time the EMJ publishes survey papers.

The EMJ publishes 4 issues in a year.

The language of the paper must be English only.

The contents of EMJ are indexed in Scopus, Web of Science (ESCI), Mathematical Reviews, MathSciNet, Zentralblatt Math (ZMATH), Referativnyi Zhurnal – Matematika, Math-Net.Ru.

The EMJ is included in the list of journals recommended by the Committee for Control of Education and Science (Ministry of Education and Science of the Republic of Kazakhstan) and in the list of journals recommended by the Higher Attestation Commission (Ministry of Education and Science of the Russian Federation).

Information for the Authors

Submission. Manuscripts should be written in LaTeX and should be submitted electronically in DVI, PostScript or PDF format to the EMJ Editorial Office via e-mail ([email protected]).

When the paper is accepted, the authors will be asked to send the tex-file of the paper to the Editorial Office.

The author who submitted an article for publication will be considered as a corresponding author. Authors may nominate a member of the Editorial Board whom they consider appropriate for the article. However, assignment to that particular editor is not guaranteed.

Copyright. When the paper is accepted, the copyright is automatically transferred to the EMJ. Manuscripts are accepted for review on the understanding that the same work has not been already published (except in the form of an abstract), that it is not under consideration for publication elsewhere, and that it has been approved by all authors.

Title page. The title page should start with the title of the paper and authors’ names (no degrees). It should contain the Keywords (no more than 10), the Subject Classification (AMS Mathematics Subject Classification (2010) with primary (and secondary) subject classification codes), and the Abstract (no more than 150 words with minimal use of mathematical symbols).

Figures. Figures should be prepared in a digital form which is suitable for direct repro- duction.

References. Bibliographical references should be listed alphabetically at the end of the article. The authors should consult the Mathematical Reviews for the standard abbreviations of journals’ names.

Authors’ data. The authors’ affiliations, addresses and e-mail addresses should be placed after the References.

Proofs. The authors will receive proofs only once. The late return of proofs may result in the paper being published in a later issue.

Offprints. The authors will receive offprints in electronic form.

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Publication Ethics and Publication Malpractice

For information on Ethics in publishing and Ethical guidelines for journal publication see http://www.elsevier.com/publishingethics and http://www.elsevier.com/journal- authors/ethics.

Submission of an article to the EMJ implies that the work described has not been published previously (except in the form of an abstract or as part of a published lecture or academic thesis or as an electronic preprint, see http://www.elsevier.com/postingpolicy), that it is not under consideration for publication elsewhere, that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out, and that, if accepted, it will not be published elsewhere in the same form, in English or in any other language, including electronically without the written consent of the copyright-holder. In particular, translations into English of papers already published in another language are not accepted.

No other forms of scientific misconduct are allowed, such as plagiarism, falsifi- cation, fraudulent data, incorrect interpretation of other works, incorrect citations, etc. The EMJ follows the Code of Conduct of the Committee on Publication Ethics (COPE), and follows the COPE Flowcharts for Resolving Cases of Suspected Misconduct (http : //publicationethics.org/files/u2/New_Code.pdf). To verify originality, your article may be checked by the originality detection service CrossCheck http://www.elsevier.com/editors/plagdetect.

The authors are obliged to participate in peer review process and be ready to provide corrections, clarifications, retractions and apologies when needed. All authors of a paper should have significantly contributed to the research.

The reviewers should provide objective judgments and should point out relevant published works which are not yet cited. Reviewed articles should be treated confidentially. The reviewers will be chosen in such a way that there is no conflict of interests with respect to the research, the authors and/or the research funders.

The editors have complete responsibility and authority to reject or accept a paper, and they will only accept a paper when reasonably certain. They will preserve anonymity of reviewers and promote publication of corrections, clarifications, retractions and apologies when needed. The acceptance of a paper automatically implies the copyright transfer to the EMJ.

The Editorial Board of the EMJ will monitor and safeguard publishing ethics.

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Web-page

The web-page of EMJ is www.emj.enu.kz. One can enter the web-page by typing Eurasian Mathematical Journal in any search engine (Google, Yandex, etc.). The archive of the web- page contains all papers published in EMJ (free access).

Subscription

For Institutions

• US$ 200 (or equivalent) for one volume (4 issues)

• US$ 60 (or equivalent) for one issue For Individuals

• US$ 160 (or equivalent) for one volume (4 issues)

• US$ 50 (or equivalent) for one issue.

The price includes handling and postage.

The Subscription Form for subscribers can be obtained by e-mail:

[email protected]

The Eurasian Mathematical Journal (EMJ) The Editorial Office

The L.N. Gumilyov Eurasian National University Building no. 3

Room 306a

Tel.: +7-7172-709500 extension 33312 13 Kazhymukan St

010008 Astana Kazakhstan

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6

EMJ: from Scopus Q4 to Scopus Q3 in two years?!

Recently the list was published of all mathematical journals included in 2015 Scopus quartiles Q1 (334 journals), Q2 (318 journals), Q3 (315 journals), and Q4 (285 journals).

Altogether 1252 journals.

With great pleasure we inform our readers that the Eurasian Mathematical Journal was included in this list, currently the only mathematical journal in the Republic of Kazakhstan and Central Asia.

It was included in Q4 with the SCImago Journal &Country Rank (SJR) indicator equal to 0,101, and is somewhere at the bottom of the Q4 list. With this indicator the journal shares places from 1240 to 1248 in the list of all 2015 Scopus mathematical journals. Nevertheless, this may be considered to be a good achievement, because Scopus uses information about journals for the three previous years, i. e. for years 2013-2015, and the EMJ is in Scopus only from the first quarter of year 2015.

The SJR indicator is calculated by using a sophisticated formula, taking into account various characteristics of journals and journals publications, in particular the average number of weighted citations received in the selected year by the documents published in the selected journal in the three previous years. This formula and related comments can be viewed on the web-page

http://www.scimagojr.com/journalrank.php?category = 2601&area= 2600&page = 1&totalsize= 373

(Help/Journals/Understand tables and charts/Detailed description of SJR.)

In order to enter Q3 the SJR indicator should be greater than 0,250. It looks like the ambitious aim of entering Q3 in year 2017 is nevertheless realistic due to recognized high level of the EMJ.

We hope that all respected members of the international Editorial Board, reviewers, authors of our journal, representing more than 35 countries, and future authors will provide high quality publications in the EMJ which will allow to achieve this aim.

On behalf of the Editorial Board of the EMJ

V.I. Burenkov, E.D. Nursultanov, T.Sh. Kalmenov,

R. Oinarov, M. Otelbaev, T.V. Tararykova, A.M. Temirkhanova

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VICTOR IVANOVICH BURENKOV (to the 75th birthday)

On July 15, 2016 was the 75th birthday of Victor Ivanovich Bu- renkov, editor-in-chief of the Eurasian Mathematical Journal (together with V.A. Sadovnichy and M. Otelbaev), director of the S.M. Nikol’skii Institute of Mathematics, head of the Department of Mathematical Analysis and Theory of Functions, chairman of Dissertation Coun- cil at the RUDN University (Moscow), research fellow (part-time) at the Steklov Institute of Mathematics (Moscow), scientific supervisor of the Laboratory of Mathematical Analysis at the Russian-Armenian (Slavonic) University (Yerevan, Armenia), doctor of physical and mathematical sciences (1983), professor (1986), honorary professor of the L.N. Gumilyov Eurasian National Uni- versity (Astana, Kazakhstan, 2006), honorary doctor of the Russian-Armenian (Slavonic) University (Yerevan, Armenia, 2007), honorary member of staff of the University of Padua (Italy, 2011), honorary distinguished professor of the Cardiff School of Mathematics (UK, 2014), honorary professor of the Aktobe Regional State University (Kazakhstan, 2015).

V.I. Burenkov graduated from the Moscow Institute of Physics and Technology (1963) and completed his postgraduate studies there in 1966 under supervision of the famous Rus- sian mathematician academician S.M. Nikol’skii.

He worked at several universities, in particular for more than 10 years at the Moscow Institute of Electronics, Radio-engineering, and Automation, the RUDN University, and the Cardiff University. He also worked at the Moscow Institute of Physics and Technology, the University of Padua, and the L.N. Gumilyov Eurasian National University.

He obtained seminal scientific results in several areas of functional analysis and the theory of partial differential and integral equations. Some of his results and methods are named after him: Burenkov’s theorem of composition of absolutely continuous functions, Burenkov’s theorem on conditional hypoellipticity, Burenkov’s method of mollifiers with variable step, Burenkov’s method of extending functions, the Burenkov-Lamberti method of transition operators in the problem of spectral stability of differential operators, the Burenkov-Guliyevs conditions for boundedness of operators in Morrey-type spaces. On the whole, the results obtained by V.I. Burenkov have laid the groundwork for new perspective scientific directions in the theory of functions spaces and its applications to partial differential equations, the spectral theory in particular.

More than 30 postgraduate students from more than 10 countries gained candidate of sciences or PhD degrees under his supervision. He has published more than 170 scientific papers. The lists of his publications can be viewed on the portals MathSciNet and Math- Net.Ru. His monograph “Sobolev spaces on domains" became a popular text for both experts in the theory of function spaces and a wide range of mathematicians interested in applying the theory of Sobolev spaces.

In 2011 the conference “Operators in Morrey-type Spaces and Applications”, dedicated to his 70th birthday was held at the Ahi Evran University (Kirsehir, Turkey). Proceedings of that conference were published in the EMJ 3-3 and EMJ 4-1.

The Editorial Board of the Eurasian Mathematical Journal congratulates Victor Ivanovich Burenkov on the occasion of his 75th birthday and wishes him good health and new achievements in science and teaching!

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EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879

Volume 7, Number 3 (2016), 41 – 52

A SHAPE-TOPOLOGICAL CONTROL OF VARIATIONAL INEQUALITIES V.A. Kovtunenko, G. Leugering

Communicated by P.D. Lamberti

Key words: shape-topological control, state-constrained optimization, variational inequality, singular perturbation, inhomogeneity, shape-topological derivative.

AMS Mathematics Subject Classification: 35B25, 49J40, 49Q10, 74G70.

Abstract. A shape-topological control of singularly perturbed variational inequalities is considered in the abstract framework for state-constrained optimization problems. Aiming at asymptotic analysis, singular perturbation theory is applied to the geometry-dependent objective function and results in a shape-topological derivative. This concept is illustrated analytically in a one-dimensional example problem which is controlled by an inhomogeneity posed in a domain with moving boundary.

1 Introduction

The paper aims at a shape-topological control of geometry-dependent variational inequalities.

We consider a class of objective functions J : G 7→ R which act on two geometric objects Γ and ω. In particular, we look how a perturbation of the topology of ω will influence the shape derivative of J(Γ, ω) with respect to Γ. Our example of a shape-topological control problem will refer to ω as an inhomogeneity in the given domain, and to Γ as a moving boundary of this domain.

From a mathematical viewpoint, the principal difficulty is thatΓandωenter the objective J through a state problem which is typically expressed by partial differential equations (PDEs). Moreover, we generalize the state problem to a variational inequality subject to unilateral constraints on Γ. Therefore, to get explicit formulae, we rely on asymptotic modeling of smallω. We obtain a shape-topological derivative of the objective function, and we prove its semi-analytic expression with the help of Green type functions.

For the classical methods of the shape optimization we refer to [1, 5, 22, 24], for the topology optimization to [2, 3, 6, 16], and to [4, 7, 20] for the asymptotic theory. Our mo- tivation comes from the crack problems in fracture mechanics, see e.g. [21], aimed either to arrest or amplify a moving crack. The control is realized by posing a trial inhomogeneity in a test medium. By this, we assume nonlinear crack models subject to contact conditions resulting in variational inequalities, see [9, 14]. The asymptotic methods of regular perturbations suitable for nonlinear crack problems are discussed in [8, 12, 15], and singular perturbations of cracks in [10, 11]. In [17] we investigated a nonlinear crack with respect to the shape-topological control by inhomogeneity in two dimensions.

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42 V.A. Kovtunenko, G. Leugering

In the present work, in Section 2 we state a mathematical concept of a shape-topological control for singularly perturbed variational inequalities, and we illustrate it with a one- dimensional example problem in Section 3.

2 Concept of a shape-topological control

Our construction can be outlined in the context of shape-topological differentiability, see [13, 18, 19], as follows.

For a Hilbert spaceH and its dual spaceH^?, we deal with variational inequalities of the following type:

find u⁰ −g ∈K such that hAu⁰, v−u⁰i ≥0 for all v −g ∈K, (2.1) whereg ∈H is given, the admissible setK ⊂H is convex and closed, andA :H 7→H^? is a linear pseudo-monotone operator such that the assumption

u^ε * u⁰ weakly in H as ε&0⁺ and lim inf

ε&0⁺ hAu^ε, u⁰−u^εi ≥0 implies that the following condition holds

hAu⁰, v−u⁰i ≥lim sup

ε&0⁺

hAu^ε, v−u^εi for all v−g ∈K.

For the theory of variational inequalities (2.1) with pseudo-monotone operators A and its solvability see [23]. In particular, ifA is strongly monotone operator such that

hAv,vi

kvk² ≥α >0 for v ∈H,v 6= 0 (2.2) then the Lions–Stampacchia theorem provides the unique solution to (2.1).

We consider a singularly perturbed variational inequality: findu^ε−g ∈K such that hA_εu^ε, v−u^εi ≥0 for all v−g ∈K, (2.3) where the perturbation A_ε = A+εF_ε of the operator A of (2.1) with a linear bounded operator F_ε : H 7→ H^? is such that εkF_εk = O(ε) and A_ε is a strongly monotone operator uniformly inε which means that

hAεv,vi

kvk² ≥α >0 forv ∈H, v 6= 0 and ε∈(0, ε₀). (2.4) Our consequent consideration aims at shape-topological control by means of the state- constrained optimization: find the geometry variables(ω,Γ) from a feasible setG such that

minimum

(ω,Γ)∈G J(u^(ω,Γ)) subject toΠ(u^(ω,Γ)) = min

v−g∈KΠ(v). (2.5)

In (2.5) the functionalΠ :H 7→R such that

Π(v) := h¹₂A_εv, vi

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A shape-topological control of variational inequalities 43 associates the strain energy (SE) of the state problem. Since Π is coercive by (2.4), then variational inequality (2.3) implies the first order necessary and sufficient optimality condition for the minimization ofΠ over v−g ∈K. The parameter ε∈R+ entering (2.3) serves for variation of the geometry, we will specify this setting in examples below.

The main difficulty of the state-constrained optimization is that the geometry variables enter (2.5) in a fully implicit way. Therefore, the problem of finding its optimality condition is open. Further we rely on asymptotic models as ε & 0⁺ that needs expansion of the solutionu^ε of state problem (2.3) stated below.

Theorem 2.1. For the solutions u⁰ and u^ε of variational inequalities (2.1) and (2.3), the following properties of a corrector q˜^ε∈H

u⁰+˜q^ε−g ∈K, (2.6)

u^ε−εq˜^ε−g ∈K, (2.7)

hA_εq˜^ε+F_εu⁰−R_ε, vi= 0 for all v ∈H, (2.8) with a residual R_ε∈H^? such that

εkR_εk= O(f(ε)), (2.9)

imply the asymptotic representation in H as ε&0⁺ of the form

ku^ε−u⁰ −ε˜q^εk= O(f(ε)). (2.10) Proof. Indeed, plugging the test functionsv =u^ε−ε˜q^ε in (2.1) due to (2.7) andv =u⁰+εq˜^ε in (2.3) due to (2.6), after summation of the inequality

hAεu^ε−(Aε−εFε)u⁰, u^ε−u⁰−ε˜q^εi ≤0 and equality (2.8) with v =u^ε−u⁰−εq˜^ε multiplied by −ε, that is

h−εA_εq˜^ε−εF_εu⁰+εR_ε, u^ε−u⁰−ε˜q^εi= 0, we get

hA_ε(u^ε−u⁰ −εq˜^ε) +εR_ε, u^ε−u⁰−ε˜q^εi ≤0.

Applying here the Cauchy–Schwarz inequality together with (2.4) and (2.9) implies (2.10).

We emphasize that ε˜q^ε satisfying (2.10) is not unique but defined up to o(f(ε))-terms.

A typical example of the corrector q˜^ε isq˜ ^x_ε

implying a boundary layer in homogenization theory. Moreover, the asymptotic behaviorf(ε)of the residual in (2.10) maybe different. In the subsequent example f(ε) = ε^3/2, see Theorem 3.1.

In the following section we illustrate our construction analytically for a one-dimensional problem which obeys exact solution. In order to find a representative q˜^ε, in Section 3.1 we will realize sufficient conditions (2.6)–(2.9). As an alternative to the uniform asymptotic expansion (2.10), in Section 3.2, developing variational technique based on Green functions, we obtain a local asymptotic expansion in the near-field, and this expansion is unique.

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3 Example problem in an inhomogeneous domain

For two variable parameters ε, t∈R+, we start with the description of geometry.

We define a disconnected set joining two segments x ∈ (0, ε)∪(ε, r+t) such that 0 <

r₀ < r < r₁, r₀−r < t < r₁−r, and0< ε < ε₀ < r₀. One geometric parameter εassociates the size of inhomogeneity ω_ε = (0, ε) in the domain, and the other geometric parameter t defines the position of the moving boundaryΓt={x:x=r+t}.

The inhomogeneity is represented with the help of the characteristic function such that χ^δ_(0,ε)(x) = δ for x < ε, otherwise χ^δ_(0,ε)(x) = 1 for x > ε, where δ ∈ R+ stands for a given stiffness parameter. Its two limit cases correspond to the hole as δ & 0⁺ and to the rigid inclusion as δ%+∞.

For a fixed g ∈R, the space of functions is given by

H_t:={u∈H¹(0, r+t) : u(0) = 0}, the admissible set is represented by the inequality constraint

K_t:={u∈H_t: u(r+t) +g ≥0},

hence v −g ∈ K_t implies v(r+t) ≥ 0 and v(0) = g, and variational inequality (2.3) takes the specific form:

find u^(ε,t)−g ∈K_t such that Z r+t

0

χ^δ_(0,ε)(u^(ε,t))⁰(v−u^(ε,t))⁰dx≥0 for all v−g ∈K_t. (3.1) Here and in what follows we mark the dependence of the solution on these two geometry variablesε and t.

Variational inequality (3.1) implies the boundary value problem:

−(u^(ε,t))⁰⁰(x) = 0 forx∈(0, ε)∪(ε, r+t), (3.2)

u^(ε,t)(0) =g, (3.3)

u^(ε,t)(ε⁺)−u^(ε,t)(ε⁻) = 0, (u^(ε,t))⁰(ε⁺)−δ·(u^(ε,t))⁰(ε⁻) = 0, (3.4)

u^(ε,t)(r+t)≥0, (u^(ε,t))⁰(r+t)≥0, (3.5)

(u^(ε,t))⁰(r+t)·u^(ε,t)(r+t) = 0, (3.6)

where u^(ε,t)(ε⁻) and u^(ε,t)(ε⁺) are the limit values from below and above, respectively. It is derived from (3.1) in the standard way by applying integration by parts for all v−g ∈ K_t that

− Z r+t

0

χ^δ_(0,ε)(u^(ε,t))⁰⁰(v−u^(ε,t))dx+ (u^(ε,t))⁰(r+t) v(r+t)−u^(ε,t)(r+t)

− (u^(ε,t))⁰(ε⁺)−δ(u^(ε,t))⁰(ε⁻)

· v(ε)−u^(ε,t)(ε)

≥0.

We construct the solution to (3.2)–(3.6) explicitly. Indeed, for an arbitrary c_(ε,t) ∈ R relations (3.2)–(3.4) can be solved by

u^(ε,t)(x) =g+ ^c^(ε,t)_δ x, (u^(ε,t))⁰ = ^c^(ε,t)_δ , x∈(0, ε) u^(ε,t)(x) =g+c_(ε,t) x+ε^1−δ_δ

, (u^(ε,t))⁰ =c_(ε,t) , x∈(ε, r+t)

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A shape-topological control of variational inequalities 45 implying the piecewise-linear continuous function

u^(ε,t)(x) =g +c_(ε,t) x+ ^1−δ_δ min{ε, x}

. (3.7)

With (3.7) complementarity condition (3.6) takes the form c(ε,t)· g+c(ε,t)(r+t+ ^1−δ_δ ε)

= 0.

Hence, due to (3.5), the nonnegative constant c_(ε,t) can be found uniquely:

c_(ε,t) = max

0,−g r+t+^1−δ_δ ε−1

. (3.8)

As ε&0⁺, from (3.7) and (3.8) we have the reference state

u^(0,t)(x) =g+c_(0,t)x, (3.9)

c_(0,t)= max

0,−g(r+t)⁻¹ , (3.10)

which solves the reference variational inequality corresponding to (2.1):

find u^(0,t)−g ∈K_t such that Z r+t

0

(u^(0,t))⁰(v−u^(0,t))⁰dx≥0 for all v−g ∈K_t. (3.11) Alluding to the asymptotic expansion in Theorem 3.1 below, we need to consider a layer near the interface pointx=ε. It is obtained after mapping(0, ε)7→(0,1), x7→εyby solving the auxiliary transmission problem:

find w∈H¹(R+) such that Z ∞

0

χ^δ_(0,1)w⁰(y)v⁰(y)dy= (1−δ)v(1) for all v ∈H¹(R⁺). (3.12) Using integration by parts, variational equation (3.12) implies the boundary value problem:

−w⁰⁰(y) = 0 for y∈(0,1)∪(1,∞), w(x)→0 as x% ∞,

w(1⁺)−w(1⁻) = 0, w⁰(1⁺)−δ·w⁰(1⁻) =−(1−δ),

where w(1⁻) and w(1⁺) are the limiting values from below and above, respectively. The unique solution of this problem is given by the piecewise linear continuous function

w(y) = ^1−δ_δ min{0, y−1}. (3.13)

After stretching the coordinates y= ^x_ε in (3.13), we get the boundary layer

εw(^x_ε) = ^1−δ_δ min{0, x−ε}, kεw(^x_ε)k= O(ε^1/2) in H¹(R+), (3.14) where the square root asymptotic order is due to the seminorm estimate

s Z ∞

0

εw(^x_ε)⁰2

dx = s

Z ε

0 1−δ

δ

2

dx= O(√ ε).

In this case we justify asymptotic formula (2.10) as follows.

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Theorem 3.1. The solutions u^(ε,t) and u^(0,t) of variational inequalities (3.1) and (3.11) admit the following residual estimate as ε&0⁺:

u^(ε,t) =u^(0,t)+εq˜^(ε,t)+ O(ε^3/2) in H¹(0, r+t) (3.15) with the principal asymptotic term defined in H_t by

ε˜q^(ε,t)(x) := (u^(0,t))⁰(0)·

εw(^x_ε) +ε^1−δ_δ (1− _r+t^x )

. (3.16)

Proof. Indeed, for sufficiently small ε we have(r+t)(r+t+^1−δ_δ ε)⁻¹ >0, hence from (3.8) it follows that

c_(ε,t) = 1 + _δ(r+t)^1−δ ε⁻¹

·max

0,−g(r+t)⁻¹ , and together with (3.10) this results in the expansion

c_(ε,t) =c_(0,t) 1− _δ(r+t)^1−δ ε+ O(ε)²

. (3.17)

Substituting (3.17), (3.14), and (3.9) in (3.7) we get u^(ε,t)(x) = u^(0,t)(x)−c_(0,t)x+c_(0,t) 1− _δ(r+t)^1−δ ε

x+ ^1−δ_δ ε+εw(^x_ε)

+ O(ε²) and derive iteratively the following uniform estimates:

u^(ε,t)(x) = u^(0,t)(x) + O(ε^1/2),

u^(ε,t)(x) = u^(0,t)(x) + (u^(0,t))⁰(0)·εw(^x_ε) + O(ε), u^(ε,t)(x) = u^(0,t)(x) + (u^(0,t))⁰(0)·

εw(^x_ε) +ε^1−δ_δ (1− _r+t^x )

+ O(ε^3/2),

where we have used c(0,t) = (u^(0,t))⁰(0). The latter equality enforces (3.15) with notation (3.16), thus completing the proof.

We remark that ε˜q^(ε,t)(x) in Theorem 3.1 satisfies relations (2.6)–(2.9) in Theorem 2.1 with f(ε) = ε^3/2, which can be checked straightforwardly.

3.1 Uniform asymptotic expansion in the problem

We discuss examples for various objectivesJ(u^(ε,t))subject to the optimal stateu^(ε,t). State- constrained optimization problem (2.5) takes the specific form:

minimum

(ε,t)∈(0,ε₀)×(r₀−r,r₁−r)J(u^(ε,t)) subject toΠ(u^(ε,t)) = min

v−g∈K_tΠ(v), (3.1)

and the strain energy (SE) functional Π :H_t7→R is Π(v) := ¹₂

Z r+t

0

χ^δ_(0,ε)(v⁰(x))²dx. (3.2)

Variational inequality (3.1) implies the first order optimality condition for the constrained minimization of Π over v−g ∈K_t.

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A shape-topological control of variational inequalities 47 It is important to comment that, for a fixed ε ∈ (0, ε₀), variations of the parameter t ∈ (r₀ −r, r₁ −r) describe regular perturbations of the moving boundary of the domain (0, ε)∪(ε, r+t), thus shape variation. In contrast, the limiting procedure ε & 0⁺ implies diminishing of the inhomogeneity ω_ε = (0, ε), and, hence, the topology change from the disconnected set to the 1-connected set(0, r+t).

First, we control the optimal value function J_SE = Π of strain energy (3.2) with respect to the topology change as ε & 0⁺. Relying on small ε, we substitute the optimal state u^(ε,t) with its asymptotic model (3.15) and (3.16), thus calculating the approximation of the optimal value function

Π(u^(ε,t)) = Π u^(0,t)+c_(0,t)

εw(^x_ε) +ε^1−δ_δ (1− _r+t^x )

+ O(ε^3/2)

= ¹₂ Z r+t

0

χ^δ_(0,ε) (u^(0,t))⁰+c_(0,t)

εw⁰(^x_ε)−^ε(1−δ)_δ(r+t)2

dx+ O(ε^3/2)

= ^c

2 (0,t)

2

hZ ε

0

δ 1− ^2ε(1−δ)_δ(r+t) + ^1−δ_δ2²

dx+ Z r+t

ε

1− ^2ε(1−δ)_δ(r+t) dxi

+ o(ε)

= ^c

2 (0,t)

2 (r+t− ^ε(1−δ)_δ ) + o(ε), Π(u^(0,t)) = ^c

2 (0,t)

2 (r+t),

(3.3)

due to (3.9), (3.14), and (3.2). From (3.3) it follows that the function (0, ε₀) 7→ R, ε 7→

Π(u^(ε,t)) is differentiable atε = 0 with the topological derivative

d

dεΠ(u^(ε,t))|_ε=0 =−c²_(0,t)^(1−δ)_2δ =−Π(u^(0,t))_δ(r+t)^1−δ . (3.4) Secondly, we control the objective function J_SERR = −_dt^dΠ of the strain energy release rate, which implies shape variation and associates a Griffith’s functional used in fracture mechanics.

To calculate −_dt^dΠ from (3.2), we apply the constitutive formula proven in [6]. Indeed, let a cut-off function η be such that η(x) = 0 as x < ε and η(x) = 1 as x > ε+β, with some β such that ε+β < r₀. For small s ∈ (r₀ −r−t, r₁ −r−t), the translation Φ_s : (0, r+t)7→(0, r+t+s), z =x+sη(x) yields the representation of Π(u^(ε,t+s))as

1 2

Z r+t+s

0

χ^δ_(0,ε) (u^(ε,t+s))⁰_z)²dz = ¹₂ Z r+t

0

χ^δ_(0,ε) ^(u^(ε,t+s)_1+sη^◦Φ0 ^s⁾⁰^x

2

(1 +sη⁰)dx

= Π(u^(ε,t+s)◦Φ_s)−₂^s Z r+t

0

χ^δ_(0,ε) (u^(ε,t+s)◦Φ_s)⁰2

η⁰dx+ o(s).

Since u^(ε,t+s) ◦Φ_s −g ∈ K_t, we infer u^(ε,t+s)◦ Φ_s → u^(ε,t) strongly in H_t as s → 0, and conclude, see [6] for details, with the asymptotic expansion

Π(u^(ε,t+s)) = Π(u^(ε,t))− ^s₂ Z r+t

0

χ^δ_(0,ε) (u^(ε,t))⁰2

η⁰dx+ o(s). (3.5) From (3.5) the explicit formula of the shape derivative follows directly:

J_SERR(u^(ε,t)) :=−_dt^dΠ(u^(ε,t)) = ¹₂ Z r+t

0

χ^δ_(0,ε) (u^(ε,t))⁰2

η⁰dx. (3.6)

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We observe that J_SERR depends on u^(ε,t), but not on ε˜q^(ε,t) in expansion (3.15). The latter fact is in accordance with the assertion in [18, 19].

For the shape-topological control, now we insert (3.15) in (3.6), which implies the asymptotic model

J_SERR u^(0,t)+c_(0,t)

εw(^x_ε) +ε^1−δ_δ (1− _r+t^x )

+ O(ε^3/2)

= ¹₂ Z r+t

0

χ^δ_(0,ε) (u^(0,t))⁰+c_(0,t)

εw⁰(^x_ε)− ^ε(1−δ)_δ(r+t)2

η⁰dx+ O(ε^3/2)

= ^c

2 (0,t)

2

Z ε+β

ε

1− ^2ε(1−δ)_δ(r+t)

η⁰dx+ o(ε) = ^c

2 (0,t)

2 1− ^2ε(1−δ)_δ(r+t)

+ o(ε)

=JSERR(u^(0,t))−ε c²_(0,t)_δ(r+t)^1−δ + o(ε), J_SERR(u^(0,t)) = ^c

2 (0,t)

2 .

(3.7)

In particular, (3.7) follows formula for the shape-topological derivative

d

dεJ_SERR(u^(ε,t))|_ε=0 =−c²_(0,t)_δ(r+t)^1−δ . (3.8) Moreover, in view of definition (3.6), it implies the mixed second derivative−_{∂ε ∂t}^∂² Π(u^(ε,t))|_ε=0 which is symmetric: _{∂ε ∂t}^∂² Π(u^(ε,t))|ε=0 = _{∂t ∂ε}^∂² Π(u^(ε,t))|ε=0. Thus, we have proved the following.

Theorem 3.2. For the solutions u^(ε,t) and u^(0,t) of variational inequalities (3.1)and (3.11), there exists the shape-topological derivative

d

dεJSERR(u^(ε,t))|_ε=0 =−_{∂ε ∂t}^∂² Π(u^(ε,t))|_ε=0 =−_{∂t ∂ε}^∂² Π(u^(ε,t))|_ε=0

=−c²_(0,t)_δ(r+t)^1−δ . (3.9)

3.2 Local asymptotic expansion in the problem

We recall that Theorem 3.2 is derived based on the uniform asymptotic formula (3.15) which, however, is not unique. Representation (3.15) which is uniform over domain matches the near-field (the boundary layer near inhomogeneity) and the far-field (extendable to infinity) asymptotic representations, which both are unique. This is the reason of our alternative approach to the shape-topological control. Since in one dimension the far-field is trivial (zero), here we employ only the near-field.

In the near-field of the moving boundary pointx=r+t, any solutionu^(ε,t)of homogeneous equation (3.2) can be written as a linear function

u^(ε,t)(x) =u^(ε,t)(r+t) + (u^(ε,t))⁰(r+t)·[x−(r+t)]for x > ε. (3.1) The factor in front of the principal term x−(r+t) in (3.1) is called stress intensity factor (SIF) in crack mechanics. We associate it with the objective

J_SIF(u^(ε,t)) = (u^(ε,t))⁰(r+t) =: c_(ε,t), (3.2) and we aim at proper formula for its calculation without knowledge of the analytic solution (3.7) and (3.8) from Section 3.1.

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A shape-topological control of variational inequalities 49 For this reason, we construct the Green function ζ_t (called the weight function in crack mechanics) obeying the bounded singularity ζ_t(r+t) 6= 0 and ζ_t⁰(r+t) 6= 0 at the moving boundary point x=r+t and solving the homogeneous problem:

−ζ_t⁰⁰(x) = 0 for x∈(0, r+t), (3.3)

ζt(0) = 0. (3.4)

All solutions of (3.3) and (3.4) are given by straight lines αx and defined up to arbitrary factor α6= 0. If we set the normalization condition

1 = Z r+t

0

(ζ_t⁰(x))²dx=ζ_t⁰(r+t)·ζ_t(r+t) (3.5) due to (3.3) and (3.4), then the uniqueαx satisfying (3.5) is

ζ_t(x) = ^√_r+t^x . (3.6)

Using (3.2)–(3.4) and (3.3)-(3.4), the second Green formula yields 0 =

Z r+t

0

(u^(ε,t))⁰⁰ζ_t−u^(ε,t)ζ_t⁰⁰

dx =−[[(u^(ε,t))⁰(ε)]]ζ_t(ε) +gζ_t⁰(0) + (u^(ε,t))⁰(r+t)·ζ_t(r+t)−u^(ε,t)(r+t)·ζ_t⁰(r+t),

(3.7)

where [[(u^(ε,t))⁰(ε)]] := (u^(ε,t))⁰(ε⁺)− (u^(ε,t))⁰(ε⁻) is the jump. Multiplying (3.7) either by (u^(ε,t))⁰(r+t)oru^(ε,t)(r+t)and using complementarity conditions (3.5), (3.6), we derive the representations

(u^(ε,t))⁰(r+t) = max

0, ζ_t⁰(r+t) [[(u^(ε,t))⁰(ε)]]ζ_t(ε)−gζ_t⁰(0) , (3.8) u^(ε,t)(r+t) = max

0,_ζ0 ¹

t(r+t) −[[(u^(ε,t))⁰(ε)]]ζ_t(ε) +gζ_t⁰(0) , (3.9) where we have used normalization (3.5) to get (3.8). In comparison with the explicit formula (3.8) ofc_(ε,t), expressions (3.8) and (3.9) are implicit ones. We plug in (3.8) expansion (3.15) and infer the asymptotic model

c_(ε,t):= (u^(ε,t))⁰(r+t) = max

0, ζ_t⁰(r+t) −gζ_t⁰(0)

+ (u^(0,t))⁰(0)[[w⁰(1)]]ζ_t(ε) +ζ_t(ε)O(ε) . (3.10) Moreover, we apply to (3.10) the local representation ζ_t(x) = ζ_t⁰(0)x following from (3.3) and (3.4), henceζ_t(ε) = ζ_t⁰(0)ε. In this way we have proved the following.

Theorem 3.3. For the solutions u^(ε,t) and u^(0,t) of variational inequalities (3.1)and (3.11), the following asymptotic representation of SIF holds:

J_SIF(u^(ε,t)) =c_(ε,t)= max

0, ζ_t⁰(r+t)ζ_t⁰(0) −g+ε(u^(0,t))⁰(0)[[w⁰(1)]] + O(ε²) , J_SIF(u^(0,t)) = c_(0,t)= max

0,−gζ_t⁰(r+t)ζ_t⁰(0) . (3.11)

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We note that the max-function in (3.11) is, generally, nondifferentiable with respect to ε when g = 0. Nevertheless, further we need the square of the max-function which is differentiable with respect to its argument. Indeed, the square of (3.11) constitutes the form:

c²_(ε,t)=c²_(0,t)+ 2εc_(0,t)(u^(0,t))⁰(0)[[w⁰(1)]]ζ_t⁰(r+t)ζ_t⁰(0) + O(ε²). (3.12) As the corollary of Theorem 3.3 we restate the asymptotic result on shape-topological control of J_SERR and J_SE from Section 3.1.

Inserting the exact solution (3.7) in (3.6), we get

J_SERR(u^(ε,t)) = −_dt^dΠ(u^(ε,t)) = ¹₂c²_(ε,t). (3.13) With the help of (3.12), from (3.13) we immediately obtain the shape-topological derivative

−_{∂ε ∂t}^∂² Π(u^(ε,t))|_ε=0 as

d

dεJ_SERR(u^(ε,t))|_ε=0 =c_(0,t)(u^(0,t))⁰(0)[[w⁰(1)]]ζ_t⁰(r+t)ζ_t⁰(0). (3.14) In order to validate (3.14), after substitution of the exact analytic expressions (3.9), (3.14), and (3.6) of solutions u^(0,t), w, and ζ_t, respectively, this results in _dε^dJ_SERR(u^(ε,t))|_ε=0 =

−c²_(0,t)_δ(r+t)^1−δ thus coinciding with expression (3.9) derived in Theorem 3.2.

Similarly, substituting (3.7) in Π(u^(ε,t)) given in (3.2), straightforward calculation provides equivalent expression of SE-optimal value function

J_SE(u^(ε,t)) = Π(u^(ε,t)) = ¹₂c²_(ε,t)(r+t+ ^1−δ_δ ε)

=^c²(0,t)

2 +εc_(0,t)(u^(0,t))⁰(0)[[w⁰(1)]]ζ_t⁰(r+t)ζ_t⁰(0) + O(ε²)

(r+t+^1−δ_δ ε)

= ^c

2 (0,t)

2 1− ^2(1−δ)_δ(r+t) + O(ε²)

(r+t+^1−δ_δ ε) = ^c

2 (0,t)

2 (r+t− ^1−δ_δ ε) + O(ε²),

where we have used here the expansion (3.12) of SIFc²_(ε,t). Thus, we arrive again at formula (3.3).

4 Discussion

In [17] this technique of a shape-topological control is extended to the nonlinear problem of crack–defect interaction in two dimensions, where no analytic solutions but only variational formulations are available. The semi-analytic expressions are proved for the shape- topological derivatives ofJ_SIF² and J_SERR.

Acknowledgments

V.A. Kovtunenko is supported by the Austrian Science Fund (FWF) project P26147-N26

"Object identification problems: numerical analysis", he thanks the Austrian Academy of Sciences (OeAW) and and OeAD Scientific & Technological Cooperation WTZ CZ 01/2016.

G. Leugering is supported by DFG EC 315 "Engineering of Advanced Materials".

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Victor Anatolievich Kovtunenko

Institute for Mathematics and Scientific Computing Karl-Franzens University of Graz, NAWI Graz Heinrichstr.36, 8010 Graz, Austria

and

Lavrent’ev Institute of Hydrodynamics 630090 Novosibirsk, Russia

E-mail: [email protected] G¨unter Leugering

Applied Mathematics 2

Friedrich-Alexander University of Erlangen-N¨urnberg Cauerstr.11, 91058 Erlangen, Germany

E-mail: [email protected]

Received: 14.03.2016