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Nina Nikolaevna Uraltseva Nina Nikolaevna Uraltseva (Nina Uraltseva with Mount Ararat in the background, Khor Virap, Armenia, 2004) Darya Apushkinskaya∗ Arshak Petrosyan† Henrik Shahgholian‡ Nina Nikolaevna Uraltseva was born on May 24, 1934 Nina Uraltseva graduated from school in 1951 (with in Leningrad, USSR (currently St. Petersburg, Russia), the highest distinction—a gold medal) and started her to parents Nikolai Fedorovich Uraltsev, (an engineer) study at the Faculty of Physics of LSU. She was an active and Lidiya Ivanovna Zmanovskaya (a school physics participant in an (undergraduate) student work-group teacher). Nina Uraltseva was attracted to both math- founded by Olga Aleksandrovna Ladyzhenskaya, that ematics and physics from the early stages of her life1. gave her the opportunity to further deepening into the She was a student at now famous school no. 239, then analysis of partial dierential equations (PDEs). In 1956, a school for girls, which later became specialized in she graduated from the university and the same year mathematics and physics and produced many notable married Gennady Lvovich Bir (a fellow student at the alumni. Together with her friends, Nina Uraltseva initi- Faculty of Physics). The young couple were soon blessed ated a mathematical study group at her school, under the with a son (and the only child) Kolya.2 supervision of Mikhail Birman, then a student at the Fac- During her graduate years, Uraltseva continued to be ulty of Mathematics and Mechanics of Leningrad State supervised by Olga Ladyzhenskaya. This mentorship University (LSU). In the higher grades of the school, transformed to a life-long productive collaboration and she was actively involved in the Mathematical Circle warm friendship until 2004, when Olga Ladyzhenskaya at the Palace of Young Pioneers, guided by Ilya Bakel- passed away. arXiv:2109.00658v1 [math.AP] 2 Sep 2021 man, and became a two-time winner of the city-wide Nina Uraltseva defended her Candidate of Science3 mathematical olympiad. thesis entitled “Regularity of Solutions to Multidimen- sional Quasilinear Equations and Variational Problems” ∗S.M. Nikol’skii Mathematical Institute, Peoples’ Friendship in 1960. Four years later, she became a Doctor of Sci- University of Russia (RUDN University), 117198 Moscow, Russia ence4 with a thesis “Boundary-Value Problems for Quasi- and Department of Mathematics and Computer Science, St. Pe- linear Elliptic Equations and Systems of Second Order.” tersburg State University, 199178 St. Petersburg, Russia e-mail: [email protected] 2Tragically, Kolya (Nikolai Uraltsev) passed away in heart attack †Department of Mathematics, Purdue University, West Lafayette, in 2013 (in Siegen, Germany). He was a renowned nuclear physicist, IN 47907, USA, e-mail: [email protected] author of 120 papers published in the world’s top scientic journals, ‡Department of Mathematics, KTH Royal Institute of Technology, most of them very well known internationally (with approximately 10044 Stockholm, Sweden, e-mail: [email protected] 6000 references), and two of them are in the category of renowned. 1Uraltseva’s early deceased younger brother (Igor Uraltsev) was a Kolya’s son, Gennady Uraltsev, is currently a postdoctoral fellow at famous physicist, a specialist in epsilon spectroscopy in semiconduc- the University of Virginia, working in harmonic analysis. tors. The Spin Optics Laboratory at St. Petersburg State University is 3Equivalent of Ph.D. in many countries named after him. 4Equivalent of Habilitation in many European countries 1 (the Kola Peninsula and the Kotlas area) and excavated Paleolithic ceramics. She is also a passionate lover of classical music and a regular visitor at philharmonic concerts. Mathematical Contributions Nina Uraltseva has made lasting contributions to math- ematics with her pioneering work in various directions in analysis and PDEs and the development of elegant and sophisticated analytical techniques. She is most renowned for her early work on linear and quasilinear equations of elliptic and parabolic type in collaboration with Olga Ladyzhenskaya, which is the category of clas- sics, but her contributions to the other areas such as de- generate and geometric equations, variational inequali- ties, and free boundaries are equally deep and signicant. Figure 1: Nina Uraltseva in a schoolgirl uniform, Below, we summarize Nina Uraltseva’s work with some Leningrad, 1951 details on selected results. Since 1959, she has been a member of the Chair of Math- 1 Linear and Quasilinear Equations ematical Physics at the Faculty of Mathematics and Me- 1.1 Hilbert’s 19th and 20th problems chanics of LSU (currently St. Petersburg State Univer- sity), where she became a Full Professor in 1968 and The rst three decades of Nina Uraltseva’s mathematical served as the head of the chair since 1974. career were devoted to the theory of linear and quasilin- For her fundamental contributions to the theory of ear PDEs of elliptic and parabolic type. Her rst round partial dierential equations in 1960s, Nina Uraltseva of works in 1960s, mostly in collaboration with Olga (jointly with Olga Ladyzhenskaya) was awarded the Ladyzhenskaya, was related to Hilbert’s 19th and 20th Chebyshev Prize of the Academy of Sciences of USSR problem on the existence and regularity of the minimiz- (1966) as well as one of the highest honors of the USSR, ers of the energy integrals the USSR State Prize (1969). Throughout her career, Nina Uraltseva has been an I(u) = Ê F(x, u, ∇u)dx, invited speaker in many meetings and conferences, in- Ω cluding the International Congress of Mathematicians F(x, u, p) in 1970 and 1986. In 2005, she was chosen as the Lec- where is a smooth function of its arguments Ω ℝn n ≥ 2 turer of the European Mathematical Society. and is a bounded domain in , . In her Candi- date of Science thesis, based on work [45]5, Nina Uralt- Nina Uraltseva’s mathematical achievements are seva has shown that under the assumption that F is C2, highly regarded throughout the world, and have been and satises the uniform ellipticity condition acknowledged by various awards, such as the titles of Honorary Scientist of the Russian Federation in 2000, 2 Fpipj ij ≥ mðð , m > 0, Honorary Professor of St. Petersburg State University in 2003, and Honorary Doctor of KTH Royal Institute of the minimizers u are C2, locally in Ω (i.e., on compactly Technology, Stockholm, Sweden, in 2006. In the same contained subdomains of Ω), provided they are known year, in recognition of her academic record, she received to be Lipschitz. (It has to be mentioned here that the Lip- Alexander von Humboldt Research Award. In 2017, Gov- schitz regularity of the minimizers was known from the ernment of St. Petersburg recognized her recent research earlier works of Ladyzhenskaya under natural growth by its Chebyshev Award. condition of F and its partial derivatives.) Uraltseva has Nina Uraltseva’s interests are not limited to scientic also shown that C2, regularity extends up to the bound- activities only. In her youth she used to be a very good ary )Ω under the natural requirement that both )Ω and basketball player, and an active member of the university 5 basketball team. She enjoyed hiking in the mountains, It was quite unusual at that time to base the Candidate of Science thesis on just a single paper and some of the committee members canoeing, and car driving. In the 1980s, Nina took part voiced their concerns. However, Olga Ladyzhenskaya objected deci- in ve archaeological expeditions in the north of Russia sively that it depends on the quality of the paper. 2 were trickier to treat, but already in [45], Uraltseva found a key: quadratic growth of a(x, u, p) in p-variable ða(x, u, p)ð ≤ (1 + ðpð)2, along with the corresponding conditions on the partial derivatives of a and aij in their variables. In [47], Uralt- seva proved C1, a priori bounds for solutions of (3), as wells as diagonal systems of similar type. The results in the elliptic case were further extended to the parabolic case (including systems) in a series of works of Ladyzheskaya and Uraltseva [23,24]. This extensive research, that went far beyond the orig- Figure 2: (from left to right) Nina Uraltseva, Olga La- inal scope of Hilbert’s 19th and 20th problems, was sum- dyzhenskaya, and Vladimir Smirnov in a seminar on marized in two monographs “Linear and Quasilinear mathematical physics, Leningrad, 1968 Equations of Elliptic Type” [25] (substantially enhanced in the 2nd edition [28]) and “Linear and Quasilinear Equations of Parabolic Type” [17], written in collabora- 2, uð)Ω are C . This generalized the results of Morrey in tion with Vsevolod Solonnikov; see Fig. 3. The mono- dimension n = 2 to higher dimensions. graphs became instant classics and were translated to Uraltseva’s proof was based on a deep extension of English [18,30] and other languages and have been ex- the ideas of De Giorgi and Nash for the solutions of tensively used for generations of mathematicians work- uniformly elliptic equations in divergence form with ing in elliptic and parabolic PDEs and remain so to this bounded measurable coecients, which were applica- date. ble only to the integrands of the form F(∇u). In par- ticular, one of the essential steps was to establish that 1.2 Equations with unbounded coecients v = ±uxi , i = 1, … , n, which are assumed to be bounded, satisfy the energy inequalities In a series of papers in 1979–1985, summarized in her talk at the International Congress of Mathematicians in Berkeley, CA, 1986 [57] and a survey paper with La- Ê ð∇vð22 ≤ C Ê (v − k)2ð∇ð2 dyzhenskaya [29], Uraltseva and collaborators have stud- A A k, k, ied uniformly elliptic quasilinear equations of nondiver- + CðAk,ð (1) gence type (3) and their parabolic counterparts, in the case when a and the rst derivatives of aij are possibly for all ðkð ≤ M, where Ak, is intersections of {v > k} unbounded.
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