View source for Solomon Mikhlin

{{Short description|Soviet mathematician}}
{{Use dmy dates|date=December 2020}}
{{Infobox scientist
| name              = Solomon Grigor'evich Mikhlin
| image             = Solomon_Mikhlin.jpg
| caption           = Solomon Grigor'evich Mikhlin
| birth_date        = 23 April 1908
| birth_place       = [[Cholmieč]], [[Rečyca Raion]], [[Minsk Governorate]], [[Russian Empire]]
| death_date        = {{death date and age|1990|08|29|1908|04|23|df=yes}}<ref name="Deathdate">See the section "[[Solomon Mikhlin#Death|Death]]" for a description of the circumstances and for the probable reason of discrepancies between the death date reported by different biographical sources.</ref>
| death_place       = [[Saint Petersburg]] (former [[Leningrad]])
| known_for         = {{plainlist|
*[[Elasticity theory]]
*[[singular integral]]s
*[[numerical analysis]]}}
| influenced        = 
| fields            = [[Mathematics]] and [[mechanics]]
| workplaces        = {{plainlist|
*Seismological Institute of the [[USSR Academy of Sciences]] (1932–1941)
*[[Abai University|Kazakh University]] in [[Alma Ata]] (1941–1944)
*[[Leningrad University]] (now [[Saint Petersburg State University]]) (1944–1990)
}}
| notable_students  = [[Vladimir Maz'ya]]
| prizes            = {{plainlist|
*[[Order of the Badge of Honour]] (1961)
*[[Honorary degree|Laurea honoris causa]] by the [[Chemnitz|Karl-Marx-Stadt]] [[Institute of technology|Polytechnic]] (1968)
*Membership of the [[German Academy of Sciences Leopoldina]] (1970)
*Membership of the [[Accademia Nazionale dei Lincei]] (1981)
}}
| nationality       = [[Soviet Union|Soviet]]
| alma_mater        = [[Leningrad University]] (1929)
| academic_advisors = [[Vladimir Ivanovich Smirnov (mathematician)|Vladimir Smirnov]], [[Leningrad University]], master [[Thesis or dissertation|thesis]]
| doctoral_students = see the [[Solomon Mikhlin#Teaching activity|teaching activity section]]
}}

'''Solomon Grigor'evich Mikhlin''' ({{langx|ru|link=no|Соломо́н Григо́рьевич Ми́хлин}}, real name Zalman Girshevich Mikhlin) (the [[family name]] is also [[Transliteration|transliterated]] as '''Mihlin''' or '''Michlin''') (23 April 1908 – 29 August 1990<ref name="Deathdate"/>) was a [[Soviet Union|Soviet]] [[mathematician]] of who worked in the fields of [[linear elasticity]], [[singular integral]]s and [[numerical analysis]]: he is best known for the introduction of the symbol of a [[singular integral]] operator, which eventually led to the foundation and development of the theory of [[pseudodifferential operator]]s.<ref name="symbol">According to {{Harvtxt|Fichera|1994|p=54}} and the references cited therein: see also {{harv|Maz'ya|2014|p=143}}. For more information on this subject, see the entries on [[singular integral operator]]s and on [[pseudodifferential operator]]s.</ref> 

== Biography ==
He was born in {{ill|Kholmech|ru|Холмеч}}, [[Rechytsa District]], [[Minsk Governorate]] (in present-day [[Belarus]]) on 23 April 1908; {{Harvtxt|Mikhlin|1968}} himself states in his [[Résumé|resume]] that his father was a merchant, but this assertion could be untrue since, in that period, people sometimes lied on the profession of parents in order to overcome political limitations in the access to higher education. According to a different version, his father was a [[melamed]], at a primary religious school ([[Cheder|kheder]]), and that the family was of modest means: according to the same source, Zalman was the youngest of five children.{{cn|date=June 2021}} His first wife was Victoria Isaevna Libina: Mikhlin's book {{Harv|Mikhlin|1965}} is dedicated to her memory. She died of [[peritonitis]] in 1961 during a boat trip on [[Volga]]. In 1940 they adopted a son, Grigory Zalmanovich Mikhlin, who later emigrated to  [[Haifa]], [[Israel]]. His second wife was Eugenia Yakovlevna Rubinova, born in 1918, who was his companion for the rest of his life.

===Education and academic career===
{{refimprove-section|date=June 2021}}
He graduated from a secondary school in [[Gomel]] in 1923 and entered the [[Herzen University|State Herzen Pedagogical Institute]] in 1925. {{cn|date=June 2021}} In 1927 he was transferred to the Department of Mathematics and Mechanics of [[Leningrad State University]] as a second year student, passing all the exams of the first year without attending lectures.{{cn|date=June 2021}} Among his university professors there were [[Nikolai Maximovich Günther]] and [[Vladimir Ivanovich Smirnov (mathematician)|Vladimir Ivanovich Smirnov]]. The latter became his master thesis supervisor: the topic of the thesis was the convergence of double [[Series (mathematics)|series]],<ref>A part of this thesis is probably reproduced in his paper {{Harv|Michlin|1932}}, where he thanks his master [[Vladimir Ivanovich Smirnov (mathematician)|Vladimir Ivanovich Smirnov]] but does not acknowledge him as a thesis advisor.</ref> and was defended in 1929. [[Sergei Lvovich Sobolev]] studied in the same class as Mikhlin. In 1930 he started his teaching career, working in some [[Leningrad]] institutes for short periods, as Mikhlin himself records on the document {{Harv|Mikhlin|1968}}. In 1932 he got a position at the Seismological Institute of the [[USSR Academy of Sciences]], where he worked till 1941: in 1935 he got the degree "[[Doktor nauk]]" in [[Mathematics]] and [[Physics]], without having to earn the "[[kandidat nauk]]" degree, and finally in 1937 he was promoted to the rank of professor. During World War II he became professor at the  [[Abai University|Kazakh University]] in [[Alma Ata]]. Since 1944 S.G. Mikhlin has been professor at the [[Leningrad State University]]. From 1964 to 1986 he headed the Laboratory of Numerical Methods at the Research Institute of Mathematics and Mechanics of the same university: since 1986 until his death he was a senior researcher at that laboratory.

=== Honours ===
He received the [[order of the Badge of Honour]] ({{langx|ru|link=no|Орден Знак Почёта}}) in 1961:<ref>See {{Harv|Mikhlin|1968|p=4}}.</ref> the name of the recipients of this prize was usually published in newspapers. He was awarded of the [[Honorary degree|Laurea honoris causa]] by the Karl-Marx-Stadt (now [[Chemnitz]]) [[Institute of technology|Polytechnic]] in 1968 and was elected member of the [[German Academy of Sciences Leopoldina]] in 1970 and of the [[Accademia Nazionale dei Lincei]] in 1981. As {{Harvtxt|Fichera|1994|p=51}} states, in his country he did not receive honours comparable to his scientific stature, mainly because of the racial policy of the [[communist regime]], briefly described in the following section.

==== Influence of communist antisemitism ====
He lived in one of the most difficult periods of contemporary Russian history. The state of mathematical sciences during this period is well described by {{Harvtxt|Lorentz|2002}}: [[marxist ideology]] rise in the [[USSR]] universities and [[Academia]] was one of the main themes of that period. Local administrators and [[Communist Party of the Soviet Union|communist party]] functionaries interfered with scientists on either [[Ethnic group|ethnical]] or [[ideology|ideological]] grounds.  As a matter of fact, during the war and during the creation of a new academic system, Mikhlin did not experience the same difficulties as younger [[Soviet Union|Soviet]] scientists of Jewish origin: for example he was included in the Soviet delegation in 1958, at the [[International Congress of Mathematicians]] in Edinburgh.<ref>See the report of the conference by {{Harvtxt|Aleksandrov|Kurosh|1959|p=250}}.</ref> However, {{Harvtxt|Fichera|1994|pp=56–60}}, examining the life of Mikhlin, finds it surprisingly similar to the life of [[Vito Volterra]] under the [[fascist regime]]. He notes that [[antisemitism]] in [[Communism|communist countries]] took different forms compared to his [[Nazism|nazist]] counterpart: the [[communist regime]] aimed not to the brutal [[homicide]] of Jews, but imposed on them a number of constrictions, sometimes very cruel, in order to make their life difficult. During the period from 1963 to 1981, he met Mikhlin attending several conferences in the [[Soviet Union]], and realised how he was in a state of isolation, almost marginalized inside his native community: [[Gaetano Fichera|Fichera]] describes several episodes revealing this fact.<ref>Almost all recollections of [[Gaetano Fichera]] concerning how this situation influenced his relationships with Mikhlin are presented in {{Harv|Fichera|1994|pp=56–61}}.</ref> Perhaps, the most illuminating one is the election of Mikhlin as a member of the [[Accademia Nazionale dei Lincei]]: in June 1981, Solomon G. Mikhlin was elected Foreign Member of the class of [[Mathematical sciences|mathematical]] and [[physical sciences]] of the Lincei. At first time, he was proposed as a winner of the [[Antonio Feltrinelli Prize]], but the almost sure confiscation of the prize by the [[Soviet Union|Soviet]] authorities induced the Lincei members to elect him as a member: they decided ''to honour him in a way that no political authority could alienate''.<ref>According to {{Harvtxt|Fichera|1994|p=59}}.</ref> However, Mikhlin was not allowed to visit Italy by the Soviet authorities,<ref>According to {{Harvtxt|Maz'ya|2000|p=2}}.</ref> so Fichera and his wife brought the tiny golden [[lynx]], the symbol of the Lincei membership, directly to Mikhlin's apartment in [[Leningrad]] on 17 October 1981: the only guests to that "ceremony" were [[Vladimir Maz'ya]] and his wife [[Tatyana Shaposhnikova]].
{{quote
|text = They just have power, but we have theorems. Therefore we are stronger!
|sign = Solomon G. Mikhlin 
|source = cited by {{harvs|txt|first=Vladimir|last=Maz'ya|author-link=Vladimir Maz'ya|year=2014|loc=p. 142}}
}}

=== Death ===
According to {{Harvtxt|Fichera|1994|pp=60–61}}, which refers a conversation with [[Mark Vishik]] and [[Olga Oleinik]], on 29 August 1990 Mikhlin left home to buy medicines for his wife Eugenia. On a public transport, he suffered a lethal stroke. He had no documents with him, therefore he was identified only some time after his death: this may be the cause of the difference in the death date reported on several biographies and obituary notices.<ref>See for example {{Harvtxt|Fichera|1994}} and the memorial page at the {{Harvtxt|St. Petersburg Mathematical Society|2006}}.</ref> Fichera also writes that Mikhlin's wife Eugenia survived him only a few months.

== Work ==

=== Research activity ===
He was author of [[monograph]]s and [[textbook]]s which become classics for their style. His research is devoted mainly to the following fields.<ref>Comprehensive descriptions of his work appear in the papers {{Harv|Fichera|1994}}, {{Harv|Fichera|Maz'ya|1978}} and in the references cited therein.</ref>

====Elasticity theory and boundary value problems====
In [[Elasticity theory|mathematical elasticity theory]], Mikhlin was concerned by three themes: the [[Plane elasticity|plane problem]] (mainly from 1932 to 1935), the [[theory of shells]] (from 1954) and the [[Cosserat spectrum]] (from 1967 to 1973).<ref>According to {{Harvtxt|Fichera|Maz'ya|1978|p=167}}.</ref> Dealing with the plane elasticity problem, he proposed two methods for its solution in [[multiply connected]] [[Domain (mathematical analysis)|domain]]s. The first one is based upon the so-called [[complex number|complex]] [[Green's function]] and the reduction of the related [[boundary value problem]] to [[integral equation]]s. The second method is a certain generalization of the classical [[Additive Schwarz method|Schwarz algorithm]] for the solution of the [[Dirichlet problem]] in a given domain by splitting it in simpler problems in smaller domains whose [[union (set theory)|union]] is the original one. Mikhlin studied its convergence and gave applications to special applied problems. He proved existence theorems for the fundamental problems of plane elasticity involving [[Homogeneity (physics)|inhomogeneous]] [[Isotropy|anisotropic]] [[Continuous media|media]]: these results are collected in the book {{Harv|Mikhlin|1957}}. Concerning the [[theory of shells]], there are several Mikhlin's articles dealing with it. He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called [[Stress (mechanics)|purely rotational state of stress]]. As a result of his study of this problem, Mikhlin also gave a new ([[Invariant (mathematics)|invariant]]) form of the basic equations of the theory. He also proved a theorem on [[Perturbation theory|perturbations]] of [[positive operator]]s in a [[Hilbert space]] which let him to obtain an error estimate for the problem of approximating a sloping shell by a [[Plate elasticity|plane plate]].<ref>The references pertaining to this work are {{Harv|Mikhlin|1952a}} and {{Harv|Mikhlin|1952b}}.</ref> Mikhlin studied also the [[Spectral theory|spectrum]] of the [[operator pencil]] of the classical [[Linear elasticity#Elastostatics|linear elastostatic operator]] or [[Linear elasticity#Elastostatics|Navier–Cauchy operator]]
:::<math>\boldsymbol{\mathcal{A}}(\omega)\boldsymbol{u}=\Delta_2\boldsymbol{u}+\omega\nabla\left(\nabla\cdot\boldsymbol{u}\right)</math>
where '''''<math>u</math>''''' is the [[displacement vector]], <math>\scriptstyle\Delta_2</math> is the [[vector laplacian]], <math>\scriptstyle\nabla</math> is the [[gradient]], <math>\scriptstyle\nabla\cdot</math> is the [[divergence]] and <math>\omega</math> is a [[Cosserat spectrum|Cosserat eigenvalue]]. The full description of the [[Spectral theory|spectrum]] and the proof of the [[Complete space|completeness]] of the system of [[eigenfunction]]s are also due to Mikhlin, and partly to [[Vladimir Gilelevich Maz'ya|V.G. Maz'ya]] in their only joint work.<ref>See the comprehensive survey paper of {{Harvtxt|Kozhevnikov|1999}}, describing the subject in his historical development including more recent development. The work of Mikhlin and his collaborators is summarized in the paper {{Harv|Mikhlin|1973}}: for a detailed analytical treatment, see also appendix I, pp. 271—311 of the posthumous book {{Harv|Mikhlin|Morozov|Paukshto|1995}}.</ref>

====Singular integrals and Fourier multipliers====
He is one of the founders of the [[Dimension|multi-dimensional]] theory of [[singular integral]]s, jointly with [[Francesco Tricomi]] and [[Georges Giraud]], and also one of the main contributors. By [[singular integral]] we mean an [[integral operator]] of the following form
:::<math> Au = v(\boldsymbol{x}) = \int_{\mathbb{R}^n}\frac{f(\boldsymbol{x},\boldsymbol{\theta})}{r^n}u(\boldsymbol{y})\mathrm{d}\boldsymbol{y}</math>
where <math>x \isin \mathbb{R}^n</math> is a point in the [[Dimension|''n''-dimensional]] [[euclidean space]], <math>r</math>=|'''<math>y-x</math>'''| and  <math>\scriptstyle\boldsymbol{\theta}=\frac{\boldsymbol{y}-\boldsymbol{x}}{r}</math> are the [[Hypersphere#Hyperspherical coordinates|hyperspherical coordinates]] (or the [[polar coordinates]] or the [[spherical coordinates]] respectively when <math>n=2</math> or <math>n=3</math>) of the [[point (geometry)|point]] '''<math>y</math>''' with respect to the point '''<math>x</math>'''. Such [[Operator (mathematics)|operators]] are called ''singular'' since the [[Singularity (mathematics)|singularity]] of the [[Kernel (integral operator)|kernel of the operator]] is so strong that the integral does not exist in the ordinary sense, but only in the sense of [[Cauchy principal value]].<ref>See the entry "[[Singular integral]]" for more details on this subject.</ref> Mikhlin was the first to develop a theory of [[singular integral equation]]s as a theory of [[operator equation]]s in [[function space]]s. In the papers {{Harv|Mikhlin|1936a}} and {{Harv|Mikhlin|1936b}} he found a rule for the composition of double singular integrals (i.e. in [[dimension (mathematics)|2-dimensional]] [[euclidean space]]s) and introduced the very important notion of [[symbol of a singular integral]]. This enabled him to show that the [[operator algebra|algebra of bounded singular integral operators]] is [[Isomorphism|isomorphic]] to the [[Algebra over a field|algebra]] of either [[Scalar field|scalar]] or [[Tensor field|matrix-valued function]]s. He proved the [[Fredholm's theorems]] for [[singular integral equation]]s and systems of such equations under the hypothesis of non-degeneracy of the [[symbol of a singular integral|symbol]]: he also proved that the [[Linear transformation#Index|index]] of a single singular integral equation in the [[euclidean space]] is [[zero]]. In 1961  Mikhlin developed a theory of [[Dimension (mathematics)|multidimensional]] [[singular integral equation]]s on [[Lipschitz continuity|Lipschitz spaces]]. These spaces are widely used in the theory of one-dimensional singular integral equations: however, the direct extension of the related theory to the multidimensional case meets some technical difficulties, and Mikhlin suggested another approach to this problem. Precisely, he obtained the basic properties of this kind of singular integral equations as a by-product of the  [[Lp space|L''<sup>p</sup>''-space]] theory of these equations. Mikhlin also proved<ref>See references {{Harv|Mikhlin|1956b}} and {{Harv|Mikhlin|1965|pp=225–240}}.</ref> a now classical theorem on [[Multiplier (Fourier analysis)|multipliers of Fourier transform]] in the [[Lp space|L''<sup>p</sup>''-space]], based on an analogous theorem of [[Józef Marcinkiewicz]] on [[Fourier series]]. A complete collection of his results in this field up to the 1965, as well as the contributions of other mathematicians like [[Francesco Tricomi|Tricomi]], [[Georges Giraud|Giraud]], [[Alberto Calderón|Calderón]] and [[Antoni Zygmund|Zygmund]],<ref>According to {{Harvtxt|Fichera|1994|p=52}}, Mikhlin himself (partially preceded by {{Harvtxt|Bochner|1951}}) shed light on the relationship between his theory of [[singular integral]]s and [[Calderon–Zygmund theory]], proving in the paper {{Harv|Mikhlin|1956a}} that, for [[Integral kernel|kernels]] of [[Convolution operator|convolution type]] i.e. kernels depending on the difference {{math|'''''y-x'''''}} of the two variables {{math|'''''x'''''}} and {{math|'''''y'''''}}, but not on the variable {{math|'''''x'''''}}, the [[symbol of a singular integral|symbol]] is the [[Fourier transform]] (in a generalized sense) of the kernel of the given [[singular integral operator]].</ref> is contained in the monograph {{Harv|Mikhlin|1965}}.<ref>Also the treatise {{Harv|Mikhlin|Prössdorf|1986}} contains a lot of information on this field, and an exposition of both the [[Dimension (mathematics)|one-dimensional]] and the multidimensional theory.</ref>

A synthesis of the theories of singular integrals and [[Linear operator|linear]] [[partial differential operator]]s was accomplished, in the mid 1960s, by the theory of [[pseudodifferential operator]]s: [[Joseph J. Kohn]], [[Louis Nirenberg]], [[Lars Hörmander]] and others operated this synthesis, but this theory owe his rise to the discoveries of Mikhlin, as is universally acknowledged.<ref name="symbol"/> This theory has numerous applications to [[mathematical physics]]. [[Multiplier (Fourier analysis)#Mikhlin multiplier theorem|Mikhlin's multiplier theorem]] is widely used in different branches of [[mathematical analysis]], particularly to the theory of [[differential equation]]s. The analysis of [[Fourier multiplier]]s was later forwarded by [[Lars Hörmander]], [[Walter Littman]], [[Elias Stein]], [[Charles Fefferman]] and others.

====Partial differential equations====
In four papers, published in the period 1940–1942, Mikhlin applies the [[Potential theory|potentials method]] to the [[mixed problem]] for the [[wave equation]]. In particular, he solves the mixed problem for the [[Dimension (mathematics)|two-space dimensional]] [[wave equation]] in the half [[Plane (mathematics)|plane]] by reducing it to the planar [[Abel integral equation]]. For [[Domain (mathematical analysis)|plane domains]] with a sufficiently [[Smooth function|smooth]] [[Plane curve|curvilinear]] [[Boundary (topology)|boundary]] he reduces the problem to an [[integro-differential equation]], which he is also able to solve when the boundary of the given domain is [[Analytic function|analytic]]. In 1951 Mikhlin proved the convergence of the [[Schwarz alternating method]] for second order [[elliptic partial differential equation|elliptic equation]]s.<ref>See {{Harv|Mikhlin|1951}} for further details.</ref> He also applied the methods of [[functional analysis]], at the same time as [[Mark Vishik]] but independently of him, to the investigation of [[boundary value problem]]s for degenerate second order elliptic partial differential equations.

====Numerical mathematics====
His work in this field can be divided into several branches:<ref>He is, according to {{Harvtxt|Fichera|1994|p=55}}, one of the pioneers of modern numerical analysis together with [[Boris Galerkin]], [[Alexander Ostrowski]], [[John von Neumann]], [[Walter Ritz]] and [[Mauro Picone]].</ref> in the following text, four main branches are described, and a sketch of his last researches is also given. The papers within the first branch are summarized in the monograph {{Harv|Mikhlin|1964}}, which contain the study of convergence of [[variational method]]s for problems connected with [[positive operator]]s, in particular, for some problems of [[mathematical physics]]. Both "a priori" and "a posteriori" [[Estimation theory|estimates]] of the errors concerning the [[approximation theory|approximation]] given by these methods are proved. The second branch deals with the notion of [[Numerical stability|stability of a numerical process]] introduced by Mikhlin himself. When applied to the variational method, this notion enables him to state necessary and sufficient conditions in order to minimize errors in the solution of the given problem when the error arising in the numerical construction of the [[Algebraic equations|algebraic system]] resulting from the application of the method itself is sufficiently small, no matter how large is the system's order. The third branch is the study of [[Finite difference method|variational-difference]] and [[finite element method]]s. Mikhlin studied the completeness of the [[coordinate function]]s used in this methods in the [[Sobolev space]] {{math|W<sup>1,''p''</sup>}}, deriving the [[order of approximation]] as a [[Function (mathematics)|function]] of the [[Smooth function|smoothness properties]] of the functions to be [[approximation of functions]] [[approximation theory|approximated]]. He also characterized the class of [[coordinate function]]s which give the best [[order of approximation]], and has studied the [[numerical stability|stability]] of the [[Finite difference method|variational-difference process]] and the growth of the [[condition number]] of the variation-difference [[Matrix (mathematics)|matrix]]. Mikhlin also studied the [[Finite element method|finite element]] approximation in [[Weight function|weighted]] [[Sobolev space]]s related to the numerical solution of degenerate [[Elliptic partial differential equation|elliptic equations]]. He found the optimal [[order of approximation]] for some methods of solution of [[Variational inequality|variational inequalities]]. The fourth branch of his research in [[numerical mathematics]] is a method for the solution of [[Fredholm integral equation]]s which he called ''resolvent method'': its essence rely on the possibility of substituting the [[Integral operator|kernel of the integral operator]] by its variational-difference approximation, so that the  [[Resolvent formalism|resolvent]] of the new kernel can be expressed by simple [[recurrence relation]]s. This eliminates the need to construct and solve large [[Algebraic equations|systems of equations]].<ref>See {{Harv|Mikhlin|1974}} and the references therein.</ref> During his last years, Mikhlin contributed to the [[theory of errors]] in numerical processes,<ref>See the book {{Harv|Mikhlin|1991}} and, for an overview of the contents, see also its review by {{harvtxt|Stummel|1993|pp=204–206}}.</ref> proposing the following classification of [[Numerical error|errors]].
#'''Approximation error''': is the error due to the replacement of an exact problem by an approximating one.
#'''Perturbation error''': is the error due to the inaccuracies in the computation of the data of the approximating problem.
#'''Algorithm error''': is the intrinsic error of the [[algorithm]] used for the solution of the approximating problem.
#'''[[Rounding error]]''': is the error due to the limits of [[computer arithmetic]].
This classification is useful since enables one to develop computational methods adjusted in order to diminish the errors of each particular type, following the ''[[divide et impera]]'' (divide and rule) principle.

===Teaching activity===
He was the "[[kandidat nauk]]" advisor of [[Tatyana Shaposhnikova|Tatyana O. Shaposhnikova]]. He was also [[mentor]] and friend of [[Vladimir Maz'ya]]: he was never his official [[supervisor]], but his friendship with the young undergraduate Maz'ya had a great influence on shaping his mathematical style.

==Selected publications==

===Books===
*{{Citation
|last = Mikhlin
|first = S.G.
|title = Integral equations and their applications to certain problems in mechanics, mathematical physics and technology
|place = [[Oxford]]–London–[[Edinburgh]]–New York–Paris–[[Frankfurt]]
|publisher = [[Pergamon Press]]
|year = 1957
|series = International Series of Monographs in Pure and Applied Mathematics
|volume = 5
|zbl = 0077.09903
|pages = XII+338
}}. The book of Mikhlin summarizing his results in the [[plane elasticity]] problem: according to {{Harvtxt|Fichera|1994|pp=55–56}} this is a widely known monograph in the theory of [[integral equation]]s.
*{{Citation
|last = Mikhlin
|first = S.G.
|title = Variational methods in mathematical physics
|place = [[Oxford]]–London–[[Edinburgh]]–New York–Paris–[[Frankfurt]]
|publisher = [[Pergamon Press]]
|year = 1964
|series = International Series of Monographs in Pure and Applied Mathematics
|volume = 50
|zbl = 0119.19002
|pages = XXXII+584 
}}.
*{{Citation
|last = Mikhlin
|first = S.G.
|title = Multidimensional singular integrals and integral equations
|place = [[Oxford]]–London–[[Edinburgh]]–New York–Paris–[[Frankfurt]]
|publisher = [[Pergamon Press]]
|year = 1965
|series = International Series of Monographs in Pure and Applied Mathematics
|volume = 83
|mr = 0185399
|zbl = 0129.07701
|pages = XII+255
}}. A masterpiece in the [[dimension|multidimensional]] theory of [[singular integral]]s and [[singular integral equation]]s summarizing all the results from the beginning to the year of publication, and also sketching the history of the subject.
*{{Citation
|last1 = Mikhlin
|first1 = Solomon G.
|last2 = Prössdorf
|first2 = Siegfried 
|title = Singular Integral Operators
|place = Berlin–[[Heidelberg]]–New York
|publisher = [[Springer Verlag]]
|year = 1986 
|pages = 528
|url = https://books.google.com/books?id=eaMmy99UTHgC&q=true
|mr = 0867687 
|zbl = 0612.47024
|isbn = 978-3-540-15967-4}}.
*{{Citation
|last = Mikhlin           
|first = S.G.
|title = Error analysis in numerical processes
|place = Chichester
|publisher = [[John Wiley & Sons]]
|year = 1991
|series = Pure and Applied Mathematics. A Wiley-Interscience Series of Text Monographs & Tracts
|volume = 1237
|pages = 283
|mr = 1129889 
|zbl = 0786.65038
|isbn = 978-0-471-92133-2}}. This book summarize the contributions of Mikhlin and of the former Soviet school of numerical analysis to the problem of error analysis in numerical solutions of various kind of equations: it was also reviewed by {{harvtxt|Stummel|1993|pp=204–206}} for the [[Bulletin of the American Mathematical Society]].
*{{Citation
|last1 = Mikhlin
|first1 = Solomon G.
|last2 = Morozov 
|first2 = Nikita Fedorovich
|last3 = Paukshto
|first3 = Michael V.
|title = The integral equations of the theory of elasticity
|place = [[Leipzig]]
|publisher = [[Teubner Verlag]]
|year = 1995
|series = Teubner-Texte zur Mathematik
|volume = 135
|pages = 375
|url = https://books.google.com/books?id=VWbvAAAAMAAJ
|doi = 10.1007/978-3-663-11626-4
|mr = 1314625 
|zbl = 0817.45004
|isbn = 3-8154-2060-1}}.

===Papers===
*{{citation
|last = Michlin
|first = S.G.
|title = Sur la convergence uniforme des séries de fonctions analytiques
|year = 1932
|language = fr
|journal = [[Matematicheskii Sbornik]]
|volume = 39
|issue = 3
|pages = 88–96
|url = http://mi.mathnet.ru/eng/msb/v39/i3/p88
|jfm = 58.0302.03 
|zbl = 0006.31701}}.
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = Équations intégrales singulières à deux variables indépendantes  
|journal = [[Mathematicheskii Sbornik|Recueil Mathématique (Matematicheskii Sbornik)]] |series=New Series
|language=ru
|volume = 1(43)
|issue = 4 
|pages = 535–552
|year = 1936a 
|url = http://mi.mathnet.ru/eng/msb/v43/i4/p535
|zbl = 0016.02902}}. The paper, with French title and abstract, where Solomon Mikhlin introduces the [[symbol of a singular integral operator]] as a means to calculate the composition of such kind of operators and solve [[singular integral equations]]: the integral operators considered here are defined by [[Integration (mathematics)|integration]] on the whole [[Dimension|''n''-dimensional]] (for ''n'' = 2) [[euclidean space]].
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = Complément à l'article "Équations intégrales singulières à deux variables indépendantes  
|journal = [[Mathematicheskii Sbornik|Recueil Mathématique (Matematicheskii Sbornik)]] |series=New Series
|language=ru
|volume = 1(43)
|issue = 6
|pages = 963–964
|year = 1936b 
|url = http://mi.mathnet.ru/eng/msb/v43/i6/p963
|jfm = 62.1251.02}}. In this paper, with French title and abstract, Solomon Mikhlin extends the definition of the [[symbol of a singular integral operator]] introduced before in the paper {{Harv|Mikhlin|1936a}} to integral operators defined by [[Integration (mathematics)|integration]] on a [[Dimension|(''n'' − 1)-dimensional]] [[closed manifold]] (for ''n'' = 3) in ''n''-dimensional [[euclidean space]].
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = Singular integral equations
|journal = [[Uspekhi Matematicheskikh Nauk]]
|language=ru
|volume = 3
|issue = 25
|pages = 29–112
|year = 1948
|url = http://mi.mathnet.ru/eng/umn/v3/i3/p29
|mr = 27429
}}.
*{{Citation
|last = Mikhlin
|first = S.G.
|title = On the Schwarz algorithm
|journal = [[Doklady Akademii Nauk SSSR]]
|series = novaya Seriya
|language=ru
|volume = 77
|pages = 569–571
|year = 1951
|zbl = 0054.04204
}}. 
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = An estimate of the error of approximating elastic shells by plane plates
|journal = [[Journal of Applied Mathematics and Mechanics|Prikladnaya Matematika i Mekhanika]]
|language=ru
|issue = 4
|volume = 16
|pages = 399–418
|year = 1952a
|zbl = 0048.42304
}}.
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = A theorem in operator theory and its application to the theory of elastic shells
|journal = [[Doklady Akademii Nauk SSSR]]
|language=ru
|series = novaya Seriya
|volume = 84
|pages = 909–912
|year = 1952b
|zbl = 0048.42401
}}. 
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = The theory of multidimensional singular integral equations
|journal = [[Vestnik Leningradskogo Universiteta]]
|language=ru
|volume = 11
|series = Seriya Matematika, Mekhanika, Astronomija
|issue = 1
|pages = 3–24 
|year = 1956a
|zbl = 0075.11402}}. 
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = On the multipliers of Fourier integrals
|journal = [[Doklady Akademii Nauk SSSR]]
|language=ru
|series=New Series
|volume = 109
|pages = 701–703
|year = 1956b
|zbl = 0073.08402}}. 
*{{Citation
|first = Solomon G.
|last = Mikhlin
|contribution = On Cosserat functions
|title = Probl. Mat. Analiza, kraevye Zadachi integral'nye Uravenya
|language=ru
|year = 1966
|pages = 59–69
|place = [[Leningrad]]
|zbl = 0166.37505
}}.
*{{Citation
|last = Mikhlin
|first = Solomon G.
|title = The spectrum of a family of operators in the theory of elasticity
|journal = [[Uspekhi Matematicheskikh Nauk]]
|language=ru
|volume = 28
|issue = 3(171)
|pages = 43–82
|year = 1973
|url = http://mi.mathnet.ru/eng/umn/v28/i3/p43
|mr = 415422
|zbl = 0291.35065}}
*{{Citation
|last = Mikhlin
|first = S.G.
|title = On a method for the approximate solution of integral equations
|journal = Vestn. Leningr. Univ. 
|language=ru
|volume = 13
|series = Ser. Mat. Mekh. Astron. 
|issue = 3
|pages = 26–33 
|year = 1974
|zbl = 0308.45014}}.

== See also ==
*[[Linear elasticity]]
*[[Multiplier (Fourier analysis)#Mikhlin multiplier theorem|Mikhlin multiplier theorem]]
*[[Multiplier (Fourier analysis)]]
*[[Singular integral]]s
*[[Singular integral equation]]s

==Notes==
{{Reflist|30em}}

==References==

===Biographical and general references===
*{{Citation
|last1 = Aleksandrov 
|first1 = P. S.
|author-link = Pavel Aleksandrov
|last2 = Kurosh
|first2 = A. G.
|author2-link = Aleksandr Gennadievich Kurosh
|title = International Congress of Mathematicians in Edinburg
|journal = [[Uspekhi Matematicheskikh Nauk]]
|language=ru
|volume = 14
|issue = 1(142)
|pages = 249–253
|year = 1959
|url = http://mi.mathnet.ru/eng/umn/v14/i1/p249
}}.
*{{Citation
|last1 = Babich 
|first1 = Vasilii Mikhailovich
|last2 = Bakelman
|first2 = Ilya Yakovlevich
|last3 = Koshelev 
|first3 = Alexander Ivanovich
|last4 = Maz'ya 
|first4 = Vladimir Gilelevich
|author4-link = Vladimir Maz'ya
|title = Solomon Grigor'evich Mikhlin (on the sixtieth anniversary of his birth)
|journal = [[Uspekhi Matematicheskikh Nauk]]
|volume = 23
|issue = 4(142)
|pages = 269–272
|year = 1968
|language=ru
|url = http://mi.mathnet.ru/eng/umn/v23/i4/p269
|mr = 228313 
|zbl = 0157.01202
}}.
*{{Citation
|last1 = Bakelman
|first1 = Ilya Yakovlevich
|last2 = Birman 
|first2 = Mikhail Shlemovich
|last3 = Ladyzhenskaya 
|first3 = Olga Aleksandrovna
|author3-link = Olga Aleksandrovna Ladyzhenskaya
|title = Solomon Grigor'evich Mikhlin (on the fiftieth anniversary of his birth)  
|journal = [[Uspekhi Matematicheskikh Nauk]]
|volume = 13
|issue = 5(83) 
|pages = 215–221
|year = 1958
|language=ru
|url = http://mi.mathnet.ru/eng/umn/v13/i5/p215
|zbl = 0085.00701
}}.
*{{Citation
|last1 = Dem'yanovich 
|first1 = Yuri Kazimirovich
|last2 = Il'in  
|first2 = Valentin Petrovich 
|last3 = Koshelev 
|first3 = Alexander Ivanovich
|last4 = Oleinik 
|first4 = Olga Arsen'evna  
|author4-link = Olga Oleinik
|last5 = Sobolev 
|first5 = Sergei L'vovich 
|author5-link = Sergei Sobolev
|title = Solomon Grigor'evich Mikhlin (on his eightieth birthday)
|journal = [[Uspekhi Matematicheskikh Nauk]]
|volume = 43
|issue = 4(262)
|pages = 239–240
|year = 1988 
|doi = 10.1070/RM1988v043n04ABEH001906
|bibcode = 1988RuMaS..43..249D
|language=ru
|url = http://mi.mathnet.ru/eng/umn/v43/i4/p239
|mr = 228313 
|zbl = 0157.01202
|s2cid = 250917521
}}.
*{{citation
|last = Fichera
|first = Gaetano
|author-link = Gaetano Fichera
|title = Solomon G. Mikhlin (1908–1990)
|journal = Atti della Accademia Nazionale dei Lincei, Rendiconti Lincei, Matematica e Applicazioni
|volume = 5
|series = Serie XI
|issue = 1
|language = it
|year = 1994
|zbl = 0852.01034
|pages = 49–61
}}. A detailed commemorative paper, referencing the works {{Harvtxt|Bakelman|Birman|Ladyzhenskaya|1958}}, {{Harvtxt|Babich|Bakelman|Koshelev|Maz'ya|1968}} and of {{Harvtxt|Dem'yanovich|Il'in|Koshelev|Oleinik|1988}} for the bibliographical details.
*{{Citation
|last1 = Fichera
|first1 = G.
|author-link = Gaetano Fichera
|last2 = Maz'ya
|first2 = V.
|author2-link = Vladimir Maz'ya
|title = In honor of professor Solomon G. Mikhlin on the occasion of his seventieth birthday
|journal = Applicable Analysis
|volume = 7
|issue = 3
|pages = 167–170 
|year = 1978
|zbl = 0378.01018
|doi = 10.1080/00036817808839188
}}. A short survey of the work of Mikhlin by a friend and his pupil: not as complete as the commemorative paper {{Harv|Fichera|1994}}, but very useful for the English speaking reader.
*{{Citation
|last1 = Kantorovich 
|first1 = Leonid Vital'evich  
|author-link = Leonid Kantorovich 
|last2 = Koshelev 
|first2 = Alexander Ivanovich
|last3 = Oleinik 
|first3 = Olga Arsen'evna  
|author3-link = Olga Oleinik
|last4 = Sobolev 
|first4 = Sergei L'vovich 
|author4-link = Sergei Sobolev
|title = Solomon Grigor'evich Mikhlin (on his seventieth birthday)
|journal = [[Uspekhi Matematicheskikh Nauk]]
|volume = 33
|issue = 2(200) 
|pages = 213–216  
|year = 1978 
|doi = 10.1070/RM1978v033n02ABEH002313 
|bibcode = 1978RuMaS..33..209K 
|language=ru
|url = http://mi.mathnet.ru/eng/umn/v33/i2/p213
|mr = 495520 
|zbl = 0378.01017
|s2cid = 250776686 
}}.
*{{Citation
|last = Lorentz
|first = G.G.
|author-link = George Lorentz
|title = Mathematics and politics in the Soviet Union from 1928 to 1953
|journal = [[Journal of Approximation Theory]]
|volume = 116
|issue = 2
|pages = 169–223
|year = 2002
|doi = 10.1006/jath.2002.3670 
|mr = 1911079 
|zbl = 1006.01009|doi-access = free
}}. See also the [http://www.math.ohio-state.edu/AT/LORENTZ/JAT02-0001_final.pdf final version] available from the "'''George Lorentz'''" section of the [http://www.math.ohio-state.edu/AT/ Approximation Theory web page] at the Mathematics Department of the [[Ohio State University]] (retrieved on 25 October 2009).
*{{Citation
|last = Maz'ya
|first = Vladimir
|author-link = Vladimir Maz'ya
|title = Problemi attuali dell'analisi e della fisica matematica. Atti del II simposio internazionale (Taormina, 15–17 ottobre 1998). Dedicato alla memoria del Prof. Gaetano Fichera.
|editor-last = Ricci
|editor-first = Paolo Emilio
|contribution = In memory of Gaetano Fichera
|contribution-url = http://www.aracneeditrice.it/pdf/264.pdf
|year = 2000
|pages = 1–4
|place = [[Rome|Roma]]
|publisher = Aracne Editrice
|zbl = 0977.01027
|title-link = Taormina
}}. Some vivid recollection about Gaetano Fichera by his colleague and friend [[Vladimir Gilelevich Maz'ya]]: there is a short description of the "[[ceremony]]" for the election of Mikhlin as a foreign member of the [[Accademia Nazionale dei Lincei]].
*{{Citation
|last = Maz'ya
|first = Vladimir G.
|author-link = Vladimir Maz'ya
|title = Differential equations of my young years
|place = [[Basel]]
|publisher = [[Birkhäuser Verlag]]
|year = 2014
|pages = xiii+191
|isbn = 978-3-319-01808-9
|mr =3288312
|zbl = 1303.01002
}}.
*[[:ru:Михлин, Соломон Григорьевич|Solomon Grigor'evich Mikhlin]]'s entry at the [[Russian Wikipedia]], Retrieved 28 May 2010.
*{{Citation
|last = Mikhlin
|first = Solomon G.
|script-title = ru:ЛИЧНЫЙ ЛИСТОК ПО УЧЕТУ КАДРОВ 
|trans-title = Formation record list
|publisher = [[USSR]]
|language=ru
|pages = 1–5
|date = 7 September 1968
}}. An official [[Résumé|resume]] written by Mikhlin itself to be used by the [[public authority]] in the former [[Soviet Union]]: it contains very useful (if not unique) information about his early career and school formation.

===Scientific references===
*{{Citation
|last = Bochner
|first = Salomon
|author-link = Salomon Bochner
|title = Theta Relations with Spherical Harmonics
|journal = [[PNAS]]
|issue = 12
|volume = 37
|date = 1 December 1951
|pages = 804–808
|doi =10.1073/pnas.37.12.804|pmid = 16589032
| zbl = 0044.07501|pmc = 1063475
|bibcode = 1951PNAS...37..804B
|doi-access = free
}}. 
*{{Citation
|first = Alexander  
|last = Kozhevnikov
|editor-last = Rossman 
|editor-first = Jürgen
|editor2-last = Takáč
|editor2-first = Peter 
|editor3-last = Günther 
|editor3-first = Wildenhain
|contribution = A history of the Cosserat spectrum
|contribution-url = https://books.google.com/books?id=9xPz9Mg2c_EC&q=%22A+history+of+the+Cosserat+spectrum%22+Alexander+Kozhevnikov&pg=PA223
|title = The Maz'ya anniversary collection. Vol. 1: On Maz'ya's work in functional analysis, partial differential equations and applications. Based on talks given at the conference, Rostock, Germany, August 31 – September 4, 1998
|series = Operator Theory. Advances and Applications
|volume = 109
|year = 1999
|pages = 223–234
|place = Basel
|publisher = [[Birkhäuser Verlag]]
|url = https://books.google.com/books?id=9xPz9Mg2c_EC
|zbl = 0936.35118
|isbn = 978-3-7643-6201-0
}}.
*{{citation
|last = Stummel
|first = F.
|title = Review: Error analysis in numerical processes, by Solomon G. Mikhlin
|journal = [[Bulletin of the American Mathematical Society]]
|year = 1993 
|volume = 28 
|issue = 1 
|pages = 204–206
|url = http://www.ams.org/journals/bull/1993-28-01/S0273-0979-1993-00357-4/
|doi = 10.1090/s0273-0979-1993-00357-4
|doi-access = free
}}.

== External links ==
*{{MacTutor|first=Vladimir G.|last=Maz'ya|author-link=Vladimir Maz'ya|first2=Tatyana O.|last2=Shaposhnikova|author2-link=Tatyana Shaposhnikova|first3=Daniele|last3=Tampieri|id=Mikhlin|title=Solomon Grigoryevich Mikhlin|date=March 2011}}
*{{MathGenealogy|id=34984|title=Solomon G. Mikhlin}}.
*{{Citation
|last = St. Petersburg Mathematical Society
|author-link = St. Petersburg Mathematical Society
|title = Solomon Grigor'evich Mikhlin
|year = 2006
|url = http://www.mathsoc.spb.ru/pantheon/mikhlin/index.html
|access-date = 13 November 2009}}. Memorial page at the [http://www.mathsoc.spb.ru/pantheon/ St. Petersburg Mathematical Pantheon].

{{Authority control}}

{{DEFAULTSORT:Mikhlin, Solomon}}
[[Category:1908 births]]
[[Category:1990 deaths]]
[[Category:People from Rechytsa district]]
[[Category:People from Rechitsky Uyezd]]
[[Category:Belarusian Jews]]
[[Category:Jewish scientists]]
[[Category:Mathematical analysts]]
[[Category:Mathematical physicists]]
[[Category:Soviet mathematicians]]
[[Category:Herzen University alumni]]