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# Nina Nikolaevna Uraltseva

Communicated by *Notices* Associate Editor Daniela De Silva

Nina Nikolaevna Uraltseva was born on May 24, 1934, in Leningrad, USSR (currently St. Petersburg, Russia), to parents Nikolai Fedorovich Uraltsev (an engineer) and Lidiya Ivanovna Zmanovskaya (a school physics teacher). Nina Uraltseva was attracted to both mathematics and physics from the early stages of her life.Footnote^{1} She was a student at the now famous school no. 239, then a school for girls, which later became specialized in mathematics and physics and produced many notable alumni. Together with her friends, Nina Uraltseva initiated a mathematical study group at her school, under the supervision of Mikhail Birman, then a student at the Faculty of Mathematics and Mechanics of Leningrad State University (LSU). In the higher grades of the school, she was actively involved in the Mathematical Circle at the Palace of Young Pioneers, guided by Ilya Bakelman, and became a two-time winner of the citywide mathematical olympiad.

^{1}

Uraltseva’s prematurely deceased younger brother (Igor Uraltsev) was a famous physicist, a specialist in epsilon spectroscopy in semiconductors. The Spin Optics Laboratory at St. Petersburg State University is named after him.

✖Nina Uraltseva graduated from school in 1951 (with the highest distinction—a gold medal) and started her study at the Faculty of Physics of LSU. She was an active participant in an (undergraduate) student work group founded by Olga Aleksandrovna Ladyzhenskaya, that gave her the opportunity to further deepen her study into the analysis of partial differential equations (PDEs). In 1956, she graduated from the university and the same year married Gennady Lvovich Bir (a fellow student at the Faculty of Physics). The young couple were soon blessed with a son (and only child) Kolya.Footnote^{2}

^{2}

Tragically, Kolya (Nikolai Uraltsev) passed away from a heart attack in 2013 (in Siegen, Germany). He was a renowned nuclear physicist, author of 120 papers published in the world’s top scientific journals, most of them very well known internationally (with approximately 6000 references), and two of them are in the category of renowned. Kolya’s son, Gennady Uraltsev, is currently a postdoctoral fellow at the University of Virginia, working in harmonic analysis.

✖During her graduate years, Uraltseva continued to be supervised by Olga Ladyzhenskaya. This mentorship transformed into a lifelong productive collaboration and warm friendship until 2004, when Olga Ladyzhenskaya passed away.

Nina Uraltseva defended her Candidate of ScienceFootnote^{3} thesis entitled “Regularity of solutions to multidimensional quasilinear equations and variational problems” in 1960. Four years later, she became a Doctor of ScienceFootnote^{4} with a thesis “Boundary-value problems for quasilinear elliptic equations and systems of second order.” Since 1959, she has been a member of the Chair of Mathematical Physics at the Faculty of Mathematics and Mechanics of LSU (currently St. Petersburg State University), where she became a Full Professor in 1968 and served as the head of the chair since 1974.

For her fundamental contributions to the theory of partial differential equations in the 1960s, Nina Uraltseva (jointly with Olga Ladyzhenskaya) was awarded the Chebyshev Prize of the Academy of Sciences of the USSR (1966) as well as one of the highest honors of the USSR, the USSR State Prize (1969).

Throughout her career, Nina Uraltseva has been an invited speaker at many meetings and conferences, including the International Congress of Mathematicians in 1970 and 1986. In 2005, she was chosen as the Lecturer of the European Mathematical Society.

Nina Uraltseva’s mathematical achievements are highly regarded throughout the world, and have been acknowledged by various awards, such as the titles of Honorary Scientist of the Russian Federation in 2000, Honorary Professor of St. Petersburg State University in 2003, and Honorary Doctor of KTH Royal Institute of Technology, Stockholm, Sweden, in 2006. In the same year, in recognition of her academic record, she received the Alexander von Humboldt Research Award. In 2017, the Government of St. Petersburg recognized her recent research by its Chebyshev Award.

Nina Uraltseva’s interests are not limited to scientific activities only. In her youth, she used to be a very good basketball player and an active member of the university basketball team. She enjoyed hiking in the mountains, canoeing, and driving a car. In the 1980s, Nina took part in five archaeological expeditions in the north of Russia (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 mathematics 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 classics, but her contributions to the other areas such as degenerate and geometric equations, variational inequalities, and free boundaries are equally deep and significant. Below, we summarize Nina Uraltseva’s work with some details on selected results.

## 1. Linear and Quasilinear Equations

### 1.1. Hilbert’s 19th and 20th problems

The first three decades of Nina Uraltseva’s mathematical career were devoted to the theory of linear and quasilinear PDEs of elliptic and parabolic type. Her first round of works in the 1960s, mostly in collaboration with Olga Ladyzhenskaya, was related to Hilbert’s 19th and 20th problems on the existence and regularity of the minimizers of the energy integrals

where is a smooth function of its arguments and is a bounded domain in , In her Candidate of Science thesis, based on work .17,Footnote^{5} Nina Uraltseva has shown that under the assumption that is and satisfies the uniform ellipticity condition

^{5}

In those years, it was quite unusual to base the Candidate of Science thesis on just a single paper and some of the committee members voiced their concerns. However, Olga Ladyzhenskaya objected decisively that it depends on the quality of the paper.

✖the minimizers are locally in (i.e., on compact subdomains of provided they are Lipschitz. (It has to be mentioned here that the Lipschitz regularity of the minimizers was known from the earlier works of Ladyzhenskaya under natural growth conditions on ), and its partial derivatives.) Uraltseva has also shown that regularity extends up to the boundary under the natural requirement that both and are This generalized the results of Morrey in dimension . to higher dimensions.

Uraltseva’s proof was based on a deep extension of the ideas of De Giorgi for the solutions of uniformly elliptic equations in divergence form with bounded measurable coefficients, which were applicable only to the integrands of the form In particular, one of the essential steps was to establish that . , which are assumed to be bounded, satisfy the energy inequalities ,

for all

Using similar ideas, Uraltseva was able to deduce the existence and regularity of solutions for the class of quasilinear uniformly elliptic equations in divergence form,

under natural growth conditions on

Quasilinear uniformly elliptic equations in nondivergence form,

were trickier to treat, but already in her thesis Uraltseva found a key: quadratic growth of

along with the corresponding conditions on the partial derivatives of

The results in the elliptic case were further extended to the parabolic case (including systems) in a series of works of Ladyzhenskaya and Uraltseva 9.

This extensive research, that went far beyond the original scope of Hilbert’s 19th and 20th problems, was summarized in two monographs, *Linear and Quasilinear Equations of Elliptic Type* (1964) (substantially enhanced in the 2nd edition in 1973) and *Linear and Quasilinear Equations of Parabolic Type* (1967), written in collaboration with Vsevolod Solonnikov; see Figure 3. The monographs became instant classics and were translated to English 812 and other languages and have been extensively used for generations of mathematicians working in elliptic and parabolic PDEs and remain so to this date.

### 1.2. Equations with unbounded coefficients

In a series of papers in 1979–1985, summarized in her talk at the International Congress of Mathematicians in Berkeley, CA, 1986 and a survey paper with Ladyzhenskaya 11, Uraltseva and collaborators have studied uniformly elliptic quasilinear equations of nondivergence type 3 and their parabolic counterparts, when

where

Most recent results of Nina Uraltseva in this direction are in the joint work with Alexander Nazarov 13 on the linear equations in divergence form,

and their parabolic counterparts. Their goal was to find conditions on the lower-order coefficients

the condition on

for some

## 2. Nonuniformly Elliptic and Parabolic Equations

### 2.1. Degenerate equations

Nina Uraltseva has also made a pioneering work on the regularity theory for degenerate quasilinear equations. A particular result in this direction is her 1968 proof 20 of the

or, equivalently, are the minimizers of the energy functional

The difficulty here lies in the fact that the

Uraltseva has obtained the

with scalar coefficients

with

Unfortunately, despite the utmost importance of this result, Nina Uraltseva’s proof remained unknown outside of the Soviet Union. In 1977, nine years later, it was independently reproved by Karen Uhlenbeck. Other proofs were given by Craig Evans (1982), John Lewis (1983), who extended the range of exponents to

Another work in this area that has gained the status of classic is the paper of Nina Uraltseva and Anarkul Urdaletova 25, where they proved uniform gradient estimates for bounded solutions of anisotropic degenerate equations,

under ellipticity, growth, and monotonicity conditions on the coefficients. Their results were applicable to the minimizers of the energy functional

with the exponents

### 2.2. Geometric equations

In 10, Ladyzhenskaya and Uraltseva developed a method of local a priori estimates for nonuniformly elliptic and parabolic equations, including the equations of minimal surface type,

A particular case with

In the 1990s, in a series of joint works with Vladimir Oliker (see 14 and the references therein), Nina Uraltseva studied the evolution of surfaces

with the boundary condition

among all competitors in

## 3. Variational Inequalities

Another area in which Nina Uraltseva has made significant contributions is variational inequalities, including variational problems with convex constraints that often exhibit a priori unknown sets known as free boundaries. An important example is the Signorini problem from elasticity, which describes equilibrium configurations of an elastic body resting on a rigid frictionless surface.

In a series of papers in the 1970s, as well as in the period 1986–1996, together with Arina Arkhipova, Nina Uraltseva studied elliptic and parabolic variational inequalities with unilateral and bilateral boundary constraints, known as the boundary obstacle problems, which can be viewed as scalar versions of the Signorini problem. Ultimately, these results played a fundamental role in Schumann’s proof (1989) of the

Below, we give a more detailed description of some of her most impactful results in this direction.

### 3.1. Problems with unilateral constraints

Let

In other words,

where

In turn, it is equivalent to the boundary value problem

to be understood in the appropriate weak sense, where

yet the exact sets where the first or the second equality holds are unknown. The interface

One of the theorems of Nina Uraltseva 22 states that when

for some

with a universal exponent

The idea of Uraltseva’s proof is based on an interplay between De Giorgi-type energy inequalities and the Signorini complementarity condition. Locally, near

for any

for all

holds, with

### 3.2. Diagonal systems

The results described above were extended by Arkhipova and Uraltseva 7 to the problem with two obstacles

While substantial difficulties arise near the set where

where

where

where

provided

For a more complete overview of Uraltseva’s results on variational inequalities, we refer to her own survey paper 23.

## 4. Free Boundary Problems

In the last 25 years, Uraltseva’s work has dealt with regularity issues arising in free boundary problems. She has developed powerful techniques, which have led to proving the optimal regularity results for solutions and for free boundaries. She has systematically studied how the free boundaries approach the fixed boundaries, and has developed tools to study free boundary problems for weakly coupled systems, as well as two-phase problems. The graduate textbook *Regularity of Free Boundaries in Obstacle-Type Problems* 15, written in collaboration with two of us, contains these and related results.

Some of Uraltseva’s major contributions (results, approaches) in free boundary problems are addressed below in more detail.

### 4.1. Touch between free and fixed boundary

In 3 (joint with one of us) and her follow-up paper 24, Uraltseva studied the obstacle problem close to a Dirichlet data, for smooth boundaries, where she proves that the free boundary touches the fixed boundary tangentially. The idea seemed to be inspired by related works with Oliker (see Section 2.2) and the Dam-problem in filtration.

During the potential theory program at Institute Mittag-Leffler (1999–2000) she started working on free boundary problems that originated in potential theory. Specifically, the harmonic continuation problem in potential theory, that was strongly tied to the obstacle problem, but with the lack of having a sign for the solution function. The simplest way to formulate this problem is as follows: