Vladimir Gilelevich Maz'ya (; born 31 December 1937)See .See , , and .See also . (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". and as "an outstanding mathematician of worldwide reputation",. who strongly influenced the development of mathematical analysis and the theory of partial differential equations., , ., , , .
Mazya's early achievements include: his work on Sobolev spaces, in particular the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities (1960),. his counterexamples related to Hilbert's 19th and Hilbert's 20th problem (1968),, , (p. 6, footnote 7, and p. 343 of the English translation). his solution, together with Yuri Burago, of a problem in harmonic potential theory (1967) posed by , his extension of the Wiener regularity test to –Laplacian and the proof of its sufficiency for the boundary regularity.The necessity of the condition was an open problem until 1993, when it was proved by . 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,. For a brief description of this and related researches, see . 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 in a Jewish family.See . His father died in December 1941 at the World War II front,See . and all four grandparents died during the siege of Leningrad. His mother, a state accountant,See and . chose to not remarry and dedicated her life to him: they lived on her meager salary in a 9 square meters room in a big communal apartment, shared with other four families.See , and . As a secondary school student, he repeatedly won the city's mathematics and physics olympiadsSee , and . and graduated with a gold medal.See , and .
In 1955, at the age of 18, Maz'ya entered the Mathematics and Mechanics Department of Leningrad University.See , , and . 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. However, he attracted the attention of Solomon Mikhlin who invited him at his home, thus starting their lifelong friendship: and this friendship had a great influence on him, helping him develop his mathematical style more than anyone else. According to ,Also reported by . 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".See also short accounts of their friendship in , , , and .
More details on the life of Vladimir Maz'ya, from his birth to the year 1968, can be found in his autobiography .
Maz'ya graduated from Leningrad University in 1960.See , and . The same year he gave two talks at Smirnov's seminar:According to : are less precise, simply referring of "talks" he gave, while cite only a single talk. their contents were published as a short report in the Proceedings of the USSR Academy of SciencesSee .See the books and for a complete analysis of his results. and later evolved in his "Candidate of Sciences" thesis, "Classes of sets and embedding theorems for function spaces",. See , , and : 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". which was defended in 1962.See , , and . In 1965 he earned the Doctor of Sciences degree, again from Leningrad University, defending the dissertation "Dirichlet and Neumann problems in Domains with irregular boundaries", when he was only 27.According to , , , , and : give a different year, stating that he earned the "Doctor of Sciences" degree in 1967. 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.See , , , and .
From 1960 up to 1986, he worked as a "research fellow": see , , and . at the Research Institute of Mathematics and Mechanics of Leningrad University (RIMM), being promoted from junior to senior research fellow in 1965.Precisely, he become "старший научный сотрудник", abbreviated as "ст. науч. сотр.", according to , the only source giving a precise date for this career advancement. From 1968 to 1978 he taught at the , where he was awarded the title of "professor" in 1976.See , , , and : a different version is reported by , whom state that he become professor of Applied Mathematics in 1971 but do not give any other detail about his teaching activity. From 1986 to 1990 he worked to the Leningrad Section of the of the USSR Academy of Sciences,See , , and . where he created and directed the Laboratory of Mathematical Models in Mechanics and the Consulting Center in Mathematics for Engineers.According to : states precisely that he was the chairman of the laboratory for several years, while does simply state that he was its head.
In 1978 he married Tatyana Shaposhnikova, a former doctoral student of Solomon Mikhlin, and they have a son, Michael:The only source briefly mentioning the composition of his household is . In 1990, they left the URSS for Sweden, where Prof. Maz'ya obtained the Swedish citizenship and started to work at Linköping University.See , , , and .
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.See , and .
Honors In 1962 Maz'ya was awarded the "Young Mathematician" prize by the Leningrad Mathematical Society, for his results on Sobolev spaces: he was the first winner of the prize. In 1990 he was awarded an honorary doctorate from Rostock University.See , , and . In 1999, Maz'ya received the Humboldt Prize.See . He was elected member of the Royal Society of Edinburgh in 2000,See , , and , and also the list of RSE members. and of the Swedish Academy of Science in 2002. In March 2003, he, jointly with Tatyana Shaposhnikova, was awarded the Verdaguer Prize by the French Academy of Sciences.For his work on the biography of Jacques Hadamard. See the short announcements of the . 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". 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 . He was awarded the Senior Whitehead Prize by the London Mathematical Society on 20 November 2009., (LMS 2010, p. 334): there is also the brief announcement in . In 2012 he was elected fellow of the American Mathematical Society.See the list of AMS fellows. On 30 October 2013 he was elected foreign member of the Georgian National Academy of Sciences.See his membership diploma, available from the Georgian National Academy web site.
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.See , and . 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,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). while the other, at the École Polytechnique in Paris,See also . The whole conference proceedings are published in the book (Mathematical Aspects of Boundary Element Methods 1999). was on the boundary element method. He was invited speaker at the International Mathematical Congress held in Beijing in 2002: 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,See and also the conference web site (2008). The proceedings were published under the editorship of . while the "Nordic – Russian Symposium in honour of Vladimir Maz'ya on the occasion of his 70th birthday" was held in Stockholm.See and also the conference web site (2008). On the same occasion, also a volume of the Proceedings of Symposia in Pure Mathematics was dedicated to him.See . 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.See . On the 26–31 May 2019, the international conference "Harmonic Analysis and PDE" was held in his honor at the Holon Institute of Technology.See the conference web site and also the interview .
Work
Research activity
Maz'ya authored/coauthored more than 500 publications, including 20 research monographs. Several survey articles describing his work can be found in the book , 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 Vladimir Maz'ya, Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations: https://users.mai.liu.se/vlama82/pdf/Problems2018l_0312.pdf
Theory of boundary value problems in nonsmooth domains In one of his early papers, considers the Dirichlet problem for the following linear elliptic equation:. See also for a brief remark.For a survey of this problem, including details on several contributions to its study, see .
where is a bounded region in the –dimensional euclidean space is a matrix whose first eigenvalue is not less than a fixed positive constant and whose entries are functions sufficiently smooth defined on , the closure of . , and are respectively a vector-valued function and two scalar functions sufficiently smooth on as their matrix counterpart . He proves the following a priori estimate
for the weak solution of , where is a constant depending on , , and other parameters but not depending on the moduli of continuity of the coefficients. The integrability exponents of the norms in are subject to the relations for , is an arbitrary positive number for ,