Vladimir Maz'ya
Born (1937-12-31) 31 December 1937
Leningrad, Russian SFSR
CitizenshipSweden
Alma materLeningrad University
Known for
SpouseTatyana O. Shaposhnikova
Awards
Scientific career
Institutions
WebsiteVladimir Maz'ya academic web site

Vladimir Gilelevich Maz'ya (Russian: Владимир Гилелевич Мазья; born 31 December 1937)[1][2][3] (the family name is sometimes transliterated as Mazya, Maz'ja or Mazja) is a Russian-born Swedish mathematician, hailed as "one of the most distinguished analysts of our time"[4] and as "an outstanding mathematician of worldwide reputation",[5] who strongly influenced the development of mathematical analysis and the theory of partial differential equations.[6][7]

Mazya's early achievements include: his work on Sobolev spaces, in particular the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities (1960),[8] his counterexamples related to Hilbert's 19th and Hilbert's 20th problem (1968),[9] his solution, together with Yuri Burago, of a problem in harmonic potential theory (1967) posed by Riesz & Szőkefalvi-Nagy (1955, chapter V, § 91), his extension of the Wiener regularity test to p–Laplacian and the proof of its sufficiency for the boundary regularity.[10] Maz'ya solved Vladimir Arnol'd's problem for the oblique derivative boundary value problem (1970) and Fritz John's problem on the oscillations of a fluid in the presence of an immersed body (1977).

In recent years, he proved a Wiener's type criterion for higher order elliptic equations, together with Mikhail Shubin solved a problem in the spectral theory of the Schrödinger operator formulated by Israel Gelfand in 1953,[11] found necessary and sufficient conditions for the validity of maximum principles for elliptic and parabolic systems of PDEs and introduced the so–called approximate approximations. He also contributed to the development of the theory of capacities, nonlinear potential theory, the asymptotic and qualitative theory of arbitrary order elliptic equations, the theory of ill-posed problems, the theory of boundary value problems in domains with piecewise smooth boundary.

Biography

Life and academic career

Vladimir Maz'ya was born on 31 December 1937[2] in a Jewish family.[12] His father died in December 1941 at the World War II front,[2][12][13] and all four grandparents died during the siege of Leningrad.[2][12] His mother, a state accountant,[14] chose to not remarry and dedicated her life to him:[12] they lived on her meager salary in a 9 square meters room in a big communal apartment, shared with other four families.[12][15] As a secondary school student, he repeatedly won the city's mathematics and physics olympiads[16] and graduated with a gold medal.[17]

In 1955, at the age of 18, Maz'ya entered the Mathematics and Mechanics Department of Leningrad University.[18] Taking part to the traditional mathematical olympiad of the faculty, he solved the problems for both first year and second year students and, since he did not make this a secret, the other participants did not submit their solutions causing the invalidation of the contest by the jury which therefore did not award the prize.[13] However, he attracted the attention of Solomon Mikhlin who invited him at his home, thus starting their lifelong friendship:[13] and this friendship had a great influence on him, helping him develop his mathematical style more than anyone else. According to Gohberg (1999, p. 2),[19] in the years to come, "Maz'ya was never a formal student of Mikhlin, but Mikhlin was more than a teacher for him. Maz'ya had found the topics of his dissertations by himself, while Mikhlin taught him mathematical ethics and rules of writing, referring and reviewing".[20]

More details on the life of Vladimir Maz'ya, from his birth to the year 1968, can be found in his autobiography (Maz'ya 2014).

Maz'ya graduated from Leningrad University in 1960.[1][21] The same year he gave two talks at Smirnov's seminar:[22] their contents were published as a short report in the Proceedings of the USSR Academy of Sciences[23][24] and later evolved in his "kandidat nauk" thesis, "Classes of sets and embedding theorems for function spaces",[25] which was defended in 1962.[26] In 1965 he earned the Doktor nauk degree, again from Leningrad University, defending the dissertation "Dirichlet and Neumann problems in Domains with irregular boundaries", when he was only 27.[27] Neither the first nor his second thesis were written under the guidance of an advisor: Vladimir Maz'ya never had a formal scientific adviser, choosing the research problems he worked to by himself.[28]

From 1960 up to 1986, he worked as a "research fellow"[29] at the Research Institute of Mathematics and Mechanics of Leningrad University (RIMM), being promoted from junior to senior research fellow in 1965.[30] From 1968 to 1978 he taught at the Leningrad Shipbuilding Institute, where he was awarded the title of "professor" in 1976.[31] From 1986 to 1990 he worked to the Leningrad Section of the Blagonravov Research Institute of Mechanical Engineering of the USSR Academy of Sciences,[32] where he created and directed the Laboratory of Mathematical Models in Mechanics and the Consulting Center in Mathematics for Engineers.[33]

In 1978 he married Tatyana Shaposhnikova, a former doctoral student of Solomon Mikhlin, and they have a son, Michael:[34] In 1990, they left the URSS for Sweden, where Prof. Maz'ya obtained the Swedish citizenship and started to work at Linköping University.[35]

Currently, he is honorary Senior Fellow of Liverpool University and Professor Emeritus at Linköping University: he is also member of the editorial board of several mathematical journals.[36]

Honors

In 1962 Maz'ya was awarded the "Young Mathematician" prize by the Leningrad Mathematical Society, for his results on Sobolev spaces:[25] he was the first winner of the prize.[23] In 1990 he was awarded an honorary doctorate from Rostock University.[37] In 1999, Maz'ya received the Humboldt Prize.[37][38] He was elected member of the Royal Society of Edinburgh in 2000,[39] and of the Swedish Academy of Science in 2002.[37] In March 2003, he, jointly with Tatyana Shaposhnikova, was awarded the Verdaguer Prize by the French Academy of Sciences.[40] On 31 August 2004 he was awarded the Celsius Gold Medal, the Royal Society of Sciences in Uppsala's top award, "for his outstanding research on partial differential equations and hydrodynamics".[41] He was awarded the Senior Whitehead Prize by the London Mathematical Society on 20 November 2009.[42] In 2012 he was elected fellow of the American Mathematical Society.[43] On 30 October 2013 he was elected foreign member of the Georgian National Academy of Sciences.[44]

Starting from 1993, several conferences have been held to honor him: the first one, held in that year at the University of Kyoto, was a conference on Sobolev spaces.[45] On the occasion of his 60th birthday in 1998, two international conferences were held in his honor: the one at the University of Rostock was on Sobolev spaces,[45][46] while the other, at the École Polytechnique in Paris,[45][47] was on the boundary element method. He was invited speaker at the International Mathematical Congress held in Beijing in 2002:[37] his talk is an exposition on his work on Wiener–type criteria for higher order elliptic equations. Other two conferences were held on the occasion of his 70th birthday: "Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz'ya" was held in Rome,[48] while the "Nordic – Russian Symposium in honour of Vladimir Maz'ya on the occasion of his 70th birthday" was held in Stockholm.[49] On the same occasion, also a volume of the Proceedings of Symposia in Pure Mathematics was dedicated to him.[50] On the occasion of his 80th birthday, a "Workshop on Sobolev Spaces and Partial Differential Equations" was held on 17–18 May 2018 was held at the Accademia Nazionale dei Lincei to honor him.[51] On the 26–31 May 2019, the international conference "Harmonic Analysis and PDE" was held in his honor at the Holon Institute of Technology.[52]

Work

Research activity

Because of Maz'ya's ability to give complete solutions to problems which are generally considered as unsolvable, Fichera once compared Maz'ya with Santa Rita, the 14th century Italian nun who is the Patron Saint of Impossible Causes.

— Alberto Cialdea, Flavia Lanzara and Paolo Emilio Ricci, (Cialdea, Lanzara & Ricci 2009, p. xii).

Maz'ya authored/coauthored more than 500 publications, including 20 research monographs. Several survey articles describing his work can be found in the book (Rossmann, Takáč & Wildenhain 1999a), and also the paper by Dorina and Marius Mitrea (2008) describes extensively his research achievements, so these references are the main ones in this section: in particular, the classification of the research work of Vladimir Maz'ya is the one proposed by the authors of these two references. He is also the author of Seventy (Five) Thousand Unsolved Problems in Analysis and Partial Differential Equations which collects problems he considers to be important research directions in the field [53]

Theory of boundary value problems in nonsmooth domains

In one of his early papers, Maz'ya (1961) considers the Dirichlet problem for the following linear elliptic equation:[54][55]

(1)     

where

He proves the following a priori estimate

(2)     

for the weak solution u of equation 1, where K is a constant depending on n, s, r κ and other parameters but not depending on the moduli of continuity of the coefficients. The integrability exponents of the Lp norms in Estimate 2 are subject to the relations

  1. 1/s ≥ 1/r - 2/n for n/2 > r > 1,
  2. s is an arbitrary positive number for r = n/2,

the first one of which answers positively to a conjecture proposed by Guido Stampacchia (1958, p. 237).[56]

Selected works

Papers

  • Maz'ya, Vladimir G. (1960), Классы областей и теоремы вложения функциональных пространств, Доклады Академии Наук СССР (in Russian), vol. 133, pp. 527–530, MR 0126152, Zbl 0114.31001, translated as Maz'ya, Vladimir G. (1960), "Classes of domains and imbedding theorems for function spaces", Soviet Mathematics - Doklady, vol. 1, pp. 882–885, MR 0126152, Zbl 0114.31001.
  • Maz'ya, Vladimir G. (1961), Некторые оценки решений эллиптических уравнений второго порядка, Доклады Академии Наук СССР (in Russian), vol. 137, pp. 1057–1059, Zbl 0115.08701, translated as Maz'ya, Vladimir G. (1961), "Some estimates for solutions of elliptic second-order equations", Soviet Mathematics - Doklady, vol. 2, pp. 413–415, Zbl 0115.08701.
  • Maz'ya, Vladimir G. (1968), Примеры нерегулярных решений квазилинейных эллиптических уравнений с аналитическими коэффициентами, Функциональный анализ и его приложения (in Russian), vol. 2, no. 3, pp. 53–57, MR 2020860, Zbl 0179.43601, translated in English as Maz'ya, Vladimir G. (1968), "Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients", Functional Analysis and Its Applications, 2 (3): 230–234, doi:10.1007/BF01076124, MR 2020860, S2CID 121038871, Zbl 0179.43601.
  • Maz'ya, V. G. (1969), "О слабых решениях задач Дирихле и Неймана", Труды Московского математического общества (in Russian), vol. 20, pp. 137–172, MR 0259329, Zbl 0179.43302, translated in English as Maz'ya, Vladimir G. (1971) [1969], "On weak solutions of the Dirichlet and Neumann problems", Transactions of the Moscow Mathematical Society, vol. 20, pp. 135–172, MR 0259329, Zbl 0226.35027.
  • Maz'ya, Vladimir; Shubin, Mikhail (2005), "Discreteness of spectrum and positivity criteria for Schrödinger operators", Annals of Mathematics, 162 (2): 919–942, arXiv:math/0305278, doi:10.4007/annals.2005.162.919, JSTOR 20159932, MR 2183285, S2CID 14741680, Zbl 1106.35043

Books

See also

Notes

  1. 1 2 See (Fomin & Shilov 1970, p. 824).
  2. 1 2 3 4 See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 189), (Bonnet, Sändig & Wendland 1999, p. 3) and (Mitrea & Mitrea 2008, p. vii).
  3. ↑ See also (Anolik et al. 2008, p. 287).
  4. ↑ (Mitrea & Mitrea 2008, p. viii).
  5. ↑ (Havin 2014, p. v).
  6. ↑ (Agranovich et al. 2008, p. 189), (Laptev 2010, p. v), (Chillingworth 2010).
  7. ↑ (Bonnet, Sändig & Wendland 1999, p. 3), (Mitrea & Mitrea 2008, p. vii), (Anolik et al. 2008, p. 287), (Movchan et al. 2015, p. 273).
  8. ↑ (Maz'ya 1960).
  9. ↑ (Maz'ya 1968), (Giaquinta 1983, p. 59), (Giusti 1994, p. 7, footnote 7, and p. 353) (p. 6, footnote 7, and p. 343 of the English translation).
  10. ↑ The necessity of the condition was an open problem until 1993, when it was proved by Kilpeläinen & Malý (1994).
  11. ↑ (Maz'ya & Shubin 2005). For a brief description of this and related researches, see (Mitrea & Mitrea 2008, p. xiv).
  12. 1 2 3 4 5 See (Eidus et al. 1997, p. 1).
  13. 1 2 3 See (Gohberg 1999, p. 2).
  14. ↑ See (Agranovich et al. 2003, p. 239) and (Mitrea & Mitrea 2008, p. vii).
  15. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 189) and (Mitrea & Mitrea 2008, p. viii).
  16. ↑ See (Agranovich et al. 2008, p. 189), (Bonnet, Sändig & Wendland 1999, p. 3) and (Mitrea & Mitrea 2008, p. viii).
  17. ↑ See (Agranovich et al. 2008, p. 189), (Eidus et al. 1997, p. 2) and (Mitrea & Mitrea 2008, p. viii).
  18. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 189), Bonnet, Sändig & Wendland (1999, p. 3) and (Eidus et al. 1997, p. 2).
  19. ↑ Also reported by Mitrea & Mitrea (2008, p. viii).
  20. ↑ See also short accounts of their friendship in (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 189), (Anolik et al. 2008, p. 287), (Bonnet, Sändig & Wendland 1999, p. 3) and (Eidus et al. 1997, p. 2).
  21. ↑ See (Agranovich et al. 2008, p. 189), (Anolik et al. 2008, p. 287) and (Mitrea & Mitrea 2008, p. viii).
  22. ↑ According to Agranovich et al. (2008, p. 189): Mitrea & Mitrea (2008, p. viii) are less precise, simply referring of "talks" he gave, while Anolik et al. (2008, p. 287) cite only a single talk.
  23. 1 2 See (Agranovich et al. 2008, p. 189).
  24. ↑ See the books (Maz'ja 1985) and (Maz'ya 2011) for a complete analysis of his results.
  25. 1 2 (Maz'ya 1960). See (Agranovich et al. 2008, p. 189), (Anolik et al. 2008, p. 287), (Eidus et al. 1997, p. 2) and (Mitrea & Mitrea 2008, p. viii): Agranovich et al. (2008, p. 189) refer that "In their reviews, the opponents and the external reviewer noted that the level of the work far exceeded the requirements of the Higher Certification Commission for Ph.D. theses, and his work was recognized as outstanding at the thesis defence in the Academic Council of Moscow State University".
  26. ↑ See (Agranovich et al. 2008, p. 189), (Anolik et al. 2008, p. 287), (Eidus et al. 1997, p. 2) and Mitrea & Mitrea (2008, p. viii).
  27. ↑ According to (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190), (Anolik et al. 2008, p. 287), Bonnet, Sändig & Wendland (1999, p. 3),(Eidus et al. 1997, p. 2) and Mitrea & Mitrea (2008, p. viii): Fomin & Shilov (1970, p. 824) give a different year, stating that he earned the "Doctor nauk" degree in 1967.
  28. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, pp. 189–190), (Anolik et al. 2008, p. 287), (Gohberg 1999, p. 2) and Mitrea & Mitrea (2008, p. viii).
  29. ↑ Russian: научный сотрудник: see (Agranovich et al. 2003, p. 239), (Anolik et al. 2008, p. 287), (Eidus et al. 1997, p. 2) and Mitrea & Mitrea (2008, p. viii).
  30. ↑ Precisely, he become "старший научный сотрудник", abbreviated as "ст. науч. сотр.", according to Fomin & Shilov (1970, p. 824), the only source giving a precise date for this career advancement.
  31. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190), (Anolik et al. 2008, p. 287), (Eidus et al. 1997, p. 2) and Mitrea & Mitrea (2008, p. viii): a different version is reported by Bonnet, Sändig & Wendland (1999, p. 3), whom state that he become professor of Applied Mathematics in 1971 but do not give any other detail about his teaching activity.
  32. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190), (Anolik et al. 2008, p. 287) and Mitrea & Mitrea (2008, pp. viii–ix).
  33. ↑ According to (Agranovich et al. 2003, p. 239): (Agranovich et al. 2008, p. 190) states precisely that he was the chairman of the laboratory for several years, while (Anolik et al. 2008, p. 287) does simply state that he was its head.
  34. ↑ The only source briefly mentioning the composition of his household is (Bonnet, Sändig & Wendland 1999, p. 3).
  35. ↑ See (Agranovich et al. 2003, p. 239), (Anolik et al. 2008, p. 287), (Bonnet, Sändig & Wendland 1999, p. 3), (Eidus et al. 1997, p. 2) and (Mitrea & Mitrea 2008, pp. viii–ix).
  36. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190) and (Anolik et al. 2008, p. 287).
  37. 1 2 3 4 See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190), (Anolik et al. 2008, p. 287) and (Mitrea & Mitrea 2008, pp. ix).
  38. ↑ See (O'Connor & Robertson 2009).
  39. ↑ See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190), (Anolik et al. 2008, p. 287) and (Mitrea & Mitrea 2008, pp. ix), and also the list of RSE members.
  40. ↑ For his work on the biography of Jacques Hadamard. See the short announcements of the French Academy of Sciences (2009).
  41. ↑ Sundelöf (2004, p. 33) precisely states:-"Celsiusmedaljen i guld, Societetens främsta utmärkelse, har tilldelats professor Vladimir Maz'ya, Linköping, för hans framstående forskning rörande partiella differentialkvationer och hydrodynamik". See also the brief announce (AMS 2005, p. 549).
  42. ↑ (Chillingworth 2010), (LMS 2010, p. 334): there is also the brief announcement in (AMS 2009, p. 1120).
  43. ↑ See the list of AMS fellows.
  44. ↑ See his membership diploma, available from the Georgian National Academy web site.
  45. 1 2 3 See (Agranovich et al. 2003, p. 239), (Agranovich et al. 2008, p. 190) and (Mitrea & Mitrea 2008, p. ix).
  46. ↑ The conference proceedings are published in two books, The Maz'ya Anniversary Collection: Volume 1 (1999) and The Maz'ya Anniversary Collection: Volume 2 (1999).
  47. ↑ See also Bonnet, Sändig & Wendland (1999, p. 3). The whole conference proceedings are published in the book (Mathematical Aspects of Boundary Element Methods 1999).
  48. ↑ See Mitrea & Mitrea (2008, p. ix) and also the conference web site (2008). The proceedings were published under the editorship of Cialdea, Lanzara & Ricci (2009).
  49. ↑ See Mitrea & Mitrea (2008, p. ix) and also the conference web site (2008).
  50. ↑ See (Mitrea & Mitrea 2008a).
  51. ↑ See (Cianchi, Sbordone & Tesei 2018).
  52. ↑ See the conference web site (Agranovsky et al. 2019) and also the interview (Holon Institute of Technology 2019).
  53. ↑ Vladimir Maz'ya, Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations: https://users.mai.liu.se/vlama82/pdf/Problems2018l_0312.pdf
  54. ↑ (Rossmann 1999, pp. 57–58). See also (Stampacchia 1963, p. 408) for a brief remark.
  55. ↑ For a survey of this problem, including details on several contributions to its study, see (Miranda 1970, §30, pp. 121–128).
  56. ↑ Maz'ya (1961, p. 413).

References

Biographical and general references

Scientific references

Publications and conferences and dedicated to Vladimir Maz'ya

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