# Vladimir Gilelevich Maz'ya

### Quick Info

Leningrad, USSR (now St Petersburg, Russia)

**Vladimir Maz'ya**is a Russian mathematician who works in mathematical analysis and partial differential equations.

### Biography

**Vladimir Maz'ya**'s childhood was difficult. His early years were those of World War II and as a consequence he suffered many hardships. In 1941 the Germans invaded Russia and by September of that year German troops were on the outskirts of Leningrad cutting the city off from the rest of Russia. Vladimir's father was killed fighting on the front. Maz'ya was lucky to be able to leave the besieged city at this time.

After the war, Maz'ya entered the Faculty of Mathematics and Mechanics of Leningrad State University when he was eighteen years old. Gohberg [4] describes an event from his first year at university. Maz'ya:-

... had solved all the problems for both first and second year while participating in the traditional olympiad of the Faculty. As Maz'ya did not make this a secret, his fellow students all decided not to submit their solutions. An unexpected result was that the jury deemed the contest a failure and no prizes were awarded to anyone, including the winner.There was a positive side to this story, however, for S G Mikhlin, the professor of mathematics, invited the talented young Maz'ya to his home [4]:-

Maz'ya never was a formal student of Mikhlin, but Mikhlin was for him more than a teacher. Maz'ya had found the topics of his dissertations by himself, while Mikhlin taught him mathematical ethics and rules of writing, refereeing and reviewing.By the time Maz'ya graduated from Leningrad State University in 1960 he already had published a paper.

*Solution of Dirichlet's problem for an equation of elliptic type*(Russian) was published in 1959 and

*Classes of domains and imbedding theorems for function spaces*(Russian) in 1960. After graduating, Maz'ya was appointed as a Junior Research Scientist at the Mathematics and Mechanics Institute of Leningrad University. He published the two papers

*Some estimates of solutions of second-order elliptic equations*(Russian) and

*p-conductivity and theorems on imbedding certain functional spaces into a C-space*(Russian) in 1961, and then four further papers in 1962, the year in which he was awarded his Candidate degree (equivalent to a doctorate) from Moscow State University. Also in 1962 Maz'ya was awarded the prize for the best junior mathematician by the Leningrad Mathematical Society.

In 1964 Maz'ya was promoted to the post of Senior Research Scientist and in the following year he was awarded a D.Sc. in mathematics from Leningrad State University. In addition to his position at the Mathematics and Mechanics Institute of Leningrad University, Maz'ya was appointed as professor in the Department of Mathematics of the Leningrad Shipbuilding Institute in 1968. In 1986 he was appointed as Head of the Laboratory of Mathematical Models in Mechanics at Leningrad Institute of Engineering Studies which was part of the USSR Academy of Sciences. In 1990 Maz'ya moved to Sweden to become the equivalent of an associate professor in the Department of Mathematics at Linköping University. He was promoted to full professor at Linköping University three years later. In January 2004 Maz'ya was appointed as Professor of Mathematics at Ohio State University in the United States and in addition he was appointed Professor of Mathematics at Liverpool University in England in September of the same year.

Maz'ya lists his mathematical interests as: linear and non-linear PDEs; asymptotic and numerical methods for PDEs, including homogenization and boundary elements; spectral theory; harmonic analysis; approximation theory; wavelets; elasticity theory; function spaces; ill-posed problems; non-linear potential theory; fluid mechanics; and the history of mathematics. This is a remarkable list but then one has to realise that he has around 430 items in his list of publications. One feature of Maz'ya's approach is that he summarises his work on a particular theme by writing a book on the topic bringing together results appearing in various papers. These books are usually written in collaboration with other mathematicians working in the field. Let us briefly mention a few.

He published (with Ju S Burago)

*Certain questions of potential theory and function theory for regions with irregular boundaries*(Russian) in 1967. The book is in two parts, the first is on the higher-dimensional potential theory and the solution of the boundary problems for regions with irregular boundaries while the second part is on the space of functions whose derivatives are measures. In 1979 and 1981 he published his two part work

*Imbedding theorems for Sobolev spaces*and

*On the theory of Sobolev spaces*. In collaboration with S A Nazarov and B A Plamenevskii, Maz'ya published

*Asymptotic behavior of solutions of elliptic boundary value problems under singular perturbations of the domain*in 1981. Here is part of the authors own summary of the work:-

The book deals with the construction of asymptotic expansions of solutions of elliptic boundary value problems under singular perturbations of the domains (i.e. blunted angles, cones or edges, small holes, narrow slits, etc.) A general approach is suggested, its main feature being systematic application of solutions to the so-called 'limit' problems. Singularities of these solutions do not increase from one step to another. The procedure of rearrangement of discrepancies between different limit problems is essential for the method under consideration.In 1985, together with Tatyana O Shaposhnikova, Maz'ya published

The first chapter is devoted to general boundary value problems for elliptic systems in domains perturbed near conic points. The Dirichlet problem for an arbitrary order elliptic equation in a domain with a cut off tubular neighbourhood of a smooth closed submanifold is considered in the second chapter. The main term of an asymptotic expansion for a solution of the Dirichlet problem for the Laplacian in a three-dimensional domain with a narrow slit is obtained in the third chapter. The fourth chapter deals with asymptotic expansions of solutions to a quasilinear equation of the second order.

*Theory of multipliers in spaces of differentiable functions*which was based on results discovered by the authors in 1979-80 and published in a number of papers. We should remark at this point that Tatyana Shaposhnikova is Maz'ya's wife. In 1997 (with Vladimir Kozlov) Maz'ya published

*Theory of a higher-order Sturm-Liouville equation*which Eastham summarises by writing that:-

... the authors have identified a special type of higher-order analogue of the hyperbolic Sturm-Liouville equation and they have developed a coherent theory based on the Green's function. The account is concise but full and written with clarity.In the same year the same two authors in collaboration with J Rossmann published

*Elliptic boundary value problems in domains with point singularities*. Agranovich summarises the work as follows:-

It is widely known that the authors have obtained many deep results for elliptic boundary value problems in domains with singularities. Here their experience permits them to present the results in a very complete and accessible form. Without doubt, the book will be interesting for many mathematicians working with elliptic boundary problems in smooth and non-smooth domains, and it would be frequently used in any mathematical library.Another book, which puts together work appearing in papers over a number of years, is (with S V Poborchi)

*Differentiable functions on bad domains*also published in 1997. In the following year Maz'ya made a remarkable contribution to the history of mathematics when he published

*Jacques Hadamard, a universal mathematician*in collaboration with his wife Tatyana Shaposhnikova. One year later, in 1999, Maz'ya, together with Vladimir Kozlov, published

*Differential equations with operator coefficients with applications to boundary value problems for partial differential equations*. Yu V Egorov's summary states:-

The book is very well written. The exposition is very clear and detailed, illustrated by many examples of applications of the stated theory. All the proofs are complete and rely on undergraduate university courses on real and complex analysis and some basic facts of functional analysis and of the theory of partial differential equations. As a serious mathematical monograph containing many new results, the book can be widely used by students.Very high quality books by Maz'ya continue to appear, adding greatly to the mathematical literature. For example we list a few recent works without detailing the co-authors:

*Spectral problems associated with corner singularities of solutions to elliptic equations*(2000);

*Asymptotic theory of elliptic boundary value problems in singularly perturbed domains*(2000);

*Spectral problems associated with corner singularities of solutions to elliptic equations*(2001); and

*Linear water waves*(2002).

Gohberg writes in [4]:-

Whatever he writes is beautiful, his love for art, music and literature seeming to feed his mathematical aesthetic feeling.Maz'ya has received, and continues to receive, numerous awards for his remarkable contributions. We mentioned above the prize he was awarded by the Leningrad Mathematical Society early in his career. Other prizes include the Humboldt Research Prize (1999), the Verdaguer Prize of the French Academy of Sciences (2003), and the Celsius Gold Medal of the Royal Society of Sciences at Uppsala (2004). He was awarded an honorary degree by the University of Rostock in 1990. He was elected to the Royal Society of Edinburgh (2001), and the Royal Swedish Academy of Sciences (2002). Several international conferences were organised for the occasion of his 60th birthday and for his 65th birthday.

Maz'ya was seventy years old on the last day of 2007 and two conferences were organised in 2008 to honour him for his 70th birthday. From 30 June to 3 July 2008 the Conference 'Analysis, PDEs and Applications' was held in Rome, and on 25 -27 August 2008 the Nordic-Russian Symposium was held in Stockholm in his honour. In addition the American Mathematical Society published

*Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G Maz'ya's 70th Birthday*in their Proceedings of Symposia in Pure Mathematics series.

One should certainly not imagine that Maz'ya's scientific activity has decreased. Remarkably, in 2007, the year of his 70th birthday, he published three books: (with S V Poborchii)

*Imbedding and Extension Theorems for Functions in Non-Lipschitz Domains*; (with Gershon Kresin)

*Sharp Real-Part Theorems. A Unified approach*; and (with Gunther Schmidt)

*Approximate Approximations*. In 2009 he published, jointly with Tatyana O Shaposhnikova, the major volume

*Theory of Sobolev multipliers. With applications to differential and integral operators*.

On 3 July 2009 the London Mathematical Society announced that they had awarded their Senior Whitehead Prize for 2009 to Maz'ya:-

... in recognition of his contributions to the theory of differential equations.

### References (show)

- J Rossmann, P Takác and G Wildenhain (eds.),
*The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999). - Curriculum vitae of Vladimir Maz'ya, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 331-333. - J Elschner, The work of Vladimir Maz'ya on integral and pseudodifferential operators, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 35-52. - I Gohberg, Vladimir Maz'ya: friend and mathematician. Recollections, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 1-5. - L I Hedberg, On Maz'ya's work in potential theory and the theory of function spaces, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 7-16. - N G Kuznetsov and B R Vainberg, Maz'ya's works in the linear theory of water waves, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 17-34. - A B Movchan, Contributions of V G Maz'ya to analysis of singularly perturbed boundary value problems, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 201-212. - Publications of Vladimir Maz'ya, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 335-364. - J Rossmann, Contributions of V Maz'ya to the theory of boundary value problems in nonsmooth domains, in
*J Rossmann, P Takác and G Wildenhain (eds.), The Maz'ya anniversary collection*(Birkhäuser Verlag, Basel, 1999), 53-98.

### Additional Resources (show)

Other websites about Vladimir Maz'ya:

### Honours (show)

Honours awarded to Vladimir Maz'ya

Written by J J O'Connor and E F Robertson

Last Update Semtember 2009

Last Update Semtember 2009