D. E. Apushkinskaya, A. A. Arkhipova, V. M. Babich, G. S. Weiss, I. A. Ibragimov, S. V. Kislyakov, N. V. Krylov, A. A. Laptev, A. I. Nazarov, G. A. Seregin, T. A. Suslina, H. Shahgholian, “On the 90th birthday of Nina Nikolaevna Uraltseva”, Russian Math. Surveys, 79:6 (2024), 1119

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Russian Mathematical Surveys, 2024, Volume 79, Issue 6, Pages 1119–1131
DOI: https://doi.org/10.4213/rm10201e (Mi rm10201)

Mathematical Life

On the 90th birthday of Nina Nikolaevna Uraltseva

D. E. Apushkinskaya, A. A. Arkhipova, V. M. Babich, G. S. Weiss, I. A. Ibragimov, S. V. Kislyakov, N. V. Krylov, A. A. Laptev, A. I. Nazarov, G. A. Seregin, T. A. Suslina, H. Shahgholian

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DOI: https://doi.org/10.4213/rm10201e

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Document Type: Personalia

MSC: 01A70

Language: English

Original paper language: Russian

Nina Nikolaevna Uraltseva was born on 24 May 1934 in Leningrad, in the family of Nikolai Fedorovich Uraltsev and Lidiya Ivanovna Zmanovskaya, both students of the Polytechnical Institute. In the summer of 1941, when the war began, Nina was evacuated with her mother to Samarovo (now Khanty-Mansiisk), where she went to school. On returning to Leningrad in 1944 she enrolled in school¹ no. 239, from which she graduated in 1951 with a gold medal of distinction. In the high school Nina also attended a group for extended mathematical education guided by I. Ya. Bakelman at the Leningrad Palace of Young Pioneers. She won twice the Leningrad Mathematical Olympiad for Schoolchildren.

After graduating from school, Nina enrolled in the Faculty of Physics at Leningrad State University, which she graduated with distinction in 1956. This was the first graduation of the Department of Higher Mathematics and Mathematical Physics organized by academician V. I. Smirnov. Several graduates of that year (N. N. Uraltseva, L. D. Faddeev, A. A. Ansel’m, and V. L. Gurevich) developed subsequently to prominent researchers.

Uraltseva wrote her diploma thesis under the supervision of Ol’ga Aleksandrovna Ladyzhenskaya and found in her person not a mere scientific advisor, but also an older friend and a colleague. Their fruitful collaboration lasted for more than 40 years.

At the year of her graduation from university Uraltseva married a fellow student, theoretical physicist G. L. Bir. Their son Nikolai was soon born, who became subsequently a well-known physicist.²

In 1960 Uraltseva finished her postgraduate studies at Leningrad University and defended the Ph.D. thesis Regularity of solutions of multidimensional quasilinear elliptic equations and variational problems (where Ladyzhenskaya was her scientific advisor). Four year later she defended her D.Sc. thesis Boundary-value problems for second-order quasilinear elliptic equations and systems.

Since 1959 Uraltseva has been working in the Department of Mathematical Physics of the Faculty of Mathematics and Mechanics at Leningrad State University, where she was first an assistant professor, and then an associated professor, and a full professor. In 1968 she received the title of Professor and since 1974 has been the head of the department.

Uraltseva is internationally known among mathematicians as a prominent expert on partial differential equations, the author of more than 120 papers and three monographs. For volume reasons we can only briefly describe her main results.³

1. Equations in divergence form and calculus of variations

Uraltseva’s early works (carried out mostly in collaboration with Ladyzhenskaya) were related to Hilbert’s 19th and 20th problems. These problems are known to determine in many respects the direction of investigations on quasilinear elliptic equations in the 20th century. The 20th problem asked about the existence of at least one function $u(x)$ minimizing a regular functional

$\begin{equation} J(u):=\int_\Omega F(x,u,Du)\,dx \end{equation} \tag{1}$

under the Dirichlet condition

$u|_{\partial\Omega}\!=\!u_0(x)$ (here an below

$\Omega$ is a bounded domain in

$\mathbb R^n$ ,

$n\geqslant2$ ), provided that the class of functions under consideration is sufficiently wide. The functional is regular in the sense that the Euler–Lagrange equation for it is elliptic. The 19th problem claimed that all solutions of elliptic equations with analytic coefficients that possess a certain regularity are analytic.

In the paper [2], underlying her Ph.D. thesis, Uraltseva showed that, under the assumption that $F(x,u,p)$ belongs to the class ${\mathcal C}^{2,\alpha}$ and satisfies the condition of uniform ellipticity

$\begin{equation*} \sum_{i,j=1}^n F_{p_ip_j}\xi_i\xi_j\geqslant \nu|\xi|^2,\qquad \nu>0, \end{equation*} \notag$

the minimizer

$u$ belongs to

${\mathcal C}^{2,\alpha}_{\rm loc}(\Omega)$ , provided that it is Lipschitz continuous.⁴^[x]⁴Note that Ladyzhenskaya proved earlier the Lipschitz continuity of a minimizer under the so-called natural growth assumption on partial derivatives of

$F$ with respect to its arguments. She also showed that, under the natural assumptions that

$\partial\Omega\in {\mathcal C}^{2,\alpha}$ and

$u\big|_{\partial\Omega}\in {\mathcal C}^{2,\alpha}$ the minimizer is

${\mathcal C}^{2,\alpha}$ -smooth up to the boundary. Such results had previously been established by Ch. Morrey only in dimension

$n=2$ .

Using a deep generalization of E. De Giorgi’s pioneering ideas, Uraltseva also established the solvability and smoothness of solutions of the Dirichler problem for general quasilinear uniformly elliptic equations in divergence form

$\begin{equation*} \sum_{i=1}^n D_ia_{i}(x,u,Du)+a(x,u,Du)=0 \end{equation*} \notag$

under natural growth conditions on the functions

$a_i$ and

$a$ with respect to their arguments. Subsequently, she improved these results in joint papers with Ladyzhenskaya. In particular, this yielded a full solution to Hilbert’s 19th and 20th problems for second-order equations.

In [3] she considered problems with boundary condition of the first order, and also a class of diagonal elliptic systems. Furthermore, her methods made it possible to obtain (see [4]) a priori estimates in ${\mathcal C}^{1,\alpha}$ for solutions of quasilinear uniformly elliptic equations in non-divergence form

$\begin{equation} \sum_{i,j=1}^n a_{ij}(x,u,Du)D_iD_ju+a(x,u,Du)=0 \end{equation} \tag{2}$

and for diagonal systems of similar form under the Bernstein assumption on the gradient growth of the function

$a$ :

$\begin{equation} |a(x,u,p)|\leqslant C+\mu|p|^2 \end{equation} \tag{3}$

and under appropriate assumptions about the derivatives of

$a_{ij}$ and

$a$ with respect to their arguments.⁵^[x]⁵These estimate depend on the

${\mathcal C}^{\alpha}$ -norm of the solution but when the constant

$\mu$ in (3) and similar conditions is sufficiently small, this norm can be replaced by

$\max_\Omega|u|$ .

Subsequently, these results were extended by Ladyzhenskaya and Uraltseva to parabolic equations and systems.

The results on elliptic equations were summarized in the monograph [5] (whose second revised edition [6] was published in 1973). Three years later the monograph [7] (joint with V. A. Solonnikov, another student of Ladyzhenskaya) was published, which was devoted to parabolic equations. Both books were translated into English (and the first of them also into French) and soon became classics.

Fairly recently Uraltseva and her former student A. I. Nazarov considered linear elliptic and parabolic equations in divergence form

$\begin{equation*} \begin{aligned} \, -\sum_{i,j=1}^n D_i\bigl(a_{ij}(x)D_ju\bigr)+\sum_{i=1}^n b_i(x)D_iu&=0; \\ \partial_tu-\sum_{i,j=1}^n D_i\bigl(a_{ij}(x,t)D_ju\bigr)+ \sum_{i=1}^n b_i(x,t)D_iu&=0 \end{aligned} \end{equation*} \notag$

with an additional structure condition arising in some applications (in particular, in mathematical hydrodynamics):

$\begin{equation} \operatorname{div}({\mathbf b}):=\sum_{i=1}^n D_ib_i\leqslant 0\quad \text{in the sense of distributions}. \end{equation} \tag{4}$

In [8] they addressed the question as to how ‘bad’ the lower-order coefficients

$b_i$ can actually be in order that some classical properties of solutions of these equations (such as the strong maximum principle, Harnack’s inequality, and Liouville’s theorem) still hold. They showed that, provided that (4) holds the assumptions on the

$b_i$ can be relaxed significantly on the scale of Morrey spaces in comparison with the general case.

2. Equations in non-divergence form

Investigations of Ladyzhenskaya and Uraltseva in the theory of equations in non-divergence form received an impetus for further development after the breakthrough results due to N. V. Krylov and M. V. Safonov on a priori estimates for the Hölder norm of solutions of linear equations. Using the techniques at hand, they managed to quickly extend these results to quasilinear equations of the form (2) and their parabolic analogues [9]. In [10] they obtained further a priori estimates for solutions of the Dirichlet problem for such equations and proved existence theorems for them.

Subsequently, Uraltseva and her collaborators extended such techniques to equations of the form (2) and similar parabolic equations in the case when the function $a$ and first derivatives of the leading coefficients $a_{ij}$ can have unbounded singularities, for example,

$\begin{equation*} |a(x,u,p)|\leqslant \mu|p|^2+b(x)|p|+\Phi(x),\qquad b,\Phi\in L^q(\Omega), \quad q>n. \end{equation*} \notag$

A survey of results on the Dirichlet problem for equations in non-divergence form was published as [11]; it was part of Uraltseva’s lecture at the International Congress of Mathematicians in Berkeley [12].

After that Uraltseva and her students investigated in a series of papers the oblique derivative problem with boundary condition of the form

$\begin{equation*} b(x,u,Du)=0 \quad \text{or} \quad b(x,t,u,Du)=0,\qquad x\in\partial\Omega, \end{equation*} \notag$

in the elliptic and parabolic cases, respectively, which thus defines the derivative of the solution in the direction of a certain vector field (depending on the solution itself). The function

$b(x,u,p)$ (or

$b(x,t,u,p)$ ) satisfies the non-tangency condition

$\begin{equation*} \sum_{i=1}^n b_{p_i}{\mathbf n}_i\geqslant \varkappa\cdot |b_p|,\qquad \varkappa>0. \end{equation*} \notag$

3. Degenerate equations

The list of Uraltseva’s works on equations with various gradient degeneracies of ellipticity opens with the paper [13], in which she, in particular, established the famous result on the ${\mathcal C}^{1,\alpha}$ -regularity of $p$ -harmonic functions, that is, weak solutions of the equation

$\begin{equation} \Delta_pu:=\sum_{i=1}^n D_i(|Du|^{p-2}D_iu)=0. \end{equation} \tag{5}$

Quasilinear equations containing the

$p$ -Laplacian

$\Delta_p$ for

$p>2$ fail to be elliptic when the gradient

$Du$ vanishes. As is known, a solution of such an equation does not generally have second Sobolev derivatives, and the problem consisted in establishing the continuity of the first derivatives in a neighbourhood of the locus of points where

$Du=0$ .

Uraltseva’s result was in fact just a consequence of a much more general theorem on the Hölder regularity of solutions of diagonal systems

$\begin{equation*} \sum_{i,j=1}^n D_i(a_{ij}(x,\mathbf{u})D_j\mathbf{u})=\mathbf{0},\qquad \mathbf{u}\in\mathbb R^N, \end{equation*} \notag$

whose matrix of coefficients satisfies the degenerate ellipticity condition

$\begin{equation*} \nu(|\mathbf{u}|)|\xi|^2\leqslant \sum_{i,j=1}^n a_{ij}(x,\mathbf{u})\xi_i\xi_j\leqslant \mu \nu(|\mathbf{u}|)|\xi|^2, \qquad \mu\geqslant 1, \end{equation*} \notag$

in which the increasing function

$\nu(\tau)$ satisfies

$\nu(\lambda\tau)\leqslant \lambda^s\nu(\tau)$ for all

$\lambda\geqslant 1$ and some

$s>0$ .

Unfortunately, Uraltseva’s result had long remained unnoticed abroad and was discovered anew and in an extended form by K. Uhlenbeck and some other authors.

With her doctoral student A. B. Urdaletova, Uraltseva was the first to obtain [14] results on the regularity of solutions of a certain class of elliptic equations with anisotropic degeneracy. This class contains the equation

$\begin{equation*} \sum_{i=1}^n D_i(|D_iu|^{p_i-2}D_iu)=0, \qquad 1<p_1\leqslant p_2\leqslant\cdots\leqslant p_n, \end{equation*} \notag$

belonging to equations with non-standard gradient growth, which are quite popular in recent decades.

Another type of ellipticity degeneracy (as $|Du|\to\infty$ ) is common in geometric problems. Uraltseva developed a method for finding local estimates for the modulus of the gradient of a solution for a class of such equation which contains Euler’s equation for functionals of surface area type

$\begin{equation} -\sum_{i=1}^n D_i\biggl(\frac{D_iu}{\sqrt{1+|Du|^2}}\biggr)+a(x,u, Du)=0. \end{equation} \tag{6}$

Intrinsic estimates for the gradient, which generalize results due to E. Bombieri, E. De Giorgi, and M. Miranda, were obtained in [15] (jointly with Ladyzhenskaya). Uraltseva included them in her lecture at the International Congress of Mathematicians in Nice [16]. Subsequently, the authors extended this result to a class of singularly perturbed problems [17]. Estimates near the boundary for the gradient of a solution in the case of a boundary condition of capillarity type,

$\begin{equation*} \frac{\partial_{\mathbf n} u}{\sqrt{1+|Du|^2}}=\varkappa\quad\text{on}\ \ \partial\Omega, \qquad |\varkappa|<1, \end{equation*} \notag$

and the solvability of the corresponding boundary value problems were established in a series of papers, of which we mention [18] and [19]. Importantly, these results were obtained only under the assumption of a smooth boundary, with no geometric conditions like the convexity of the domain.

In the 1990s, in a series of papers (some of which were written with V. I. Oliker; see [20] and the references there) Uraltseva investigated the evolution of surfaces $S(t)$ that are the graphs of functions $u=u(x,t)$ in a bounded domain $\Omega\subset \mathbb{R}^n$ . The boundary of the surfaces $S(t)$ is assumed to be fixed, and the velocity of motion depends on the mean curvature of $S(t)$ . This evolution is described by the Dirichlet problem for a parabolic equation:

$\begin{equation*} \begin{gathered} \, \frac {\partial_t u}{\sqrt{1+|Du|^2}}= \sum_{i=1}^n D_i\bigg(\frac{D_i u}{\sqrt{1+|Du|^2}}\bigg), \qquad x\in\Omega,\quad t>0; \\ u|_{x\in \partial\Omega}=\phi(x); \qquad u(x,0)=u_0(x). \end{gathered} \end{equation*} \notag$

Previously, for domains with non-negative mean curvature of the boundary G. Huisken proved the existence of a classical solution to this problem and showed that the surfaces

$S(t)$ converge to a classical minimal surface, and Oliker and Uraltseva considered this problem without geometric assumptions about the domain. To do this, they introduced the concept of generalized solution of a parabolic problem and proved its existence and convergence as

$t\to\infty$ to a generalized solution

$\Phi$ of the stationary problem, which minimizes the area functional

$\begin{equation*} J(u):=\int_\Omega\sqrt{1+|Du|^2}+\int_{\partial\Omega}|u-\phi| \end{equation*} \notag$

in the natural class

$W^1_1(\Omega)$ . Such a solution

$\Phi$ is unique, but can differ from the Dirichlet data

$\phi$ on a ‘bad’ part of the boundary, where the boundary has a negative mean curvature. Note that the analysis of the behaviour of

$\Phi$ close to ‘contact points’ on the boundary⁶ at which it detaches from

$\phi$ was a motivation for Uraltseva’s studies of contact between free and fixed boundaries.

4. Variational inequalities

A large cycle of Uraltseva’s papers (some of which she wrote with her former student A. A. Arkhipova) relates to the theory of variational inequalities. The simplest variational inequality for an elliptic operator arises in the minimization of a regular functional $J$ on a closed convex subset ${\mathcal K}$ of a Banach space $X$ ; it has the form

$\begin{equation*} \langle J'(u),v-u\rangle \geqslant0 \quad\text{for all}\ \ v\in {\mathcal K}. \end{equation*} \notag$

For a parabolic operator the corresponding (evolutionary) variational inequality has the form

$\begin{equation*} \langle u'(t),v-u(t)\rangle+\langle J'(u(t)),v-u(t)\rangle \geqslant0 \quad\text{for all }\ \ v\in {\mathcal K} \end{equation*} \notag$

(here

$u\colon [0,T]\to X$ is an abstract function of the real variable in a suitable class).

The analysis of the regularity of generalized solutions is an important problem in the theory of variational inequalities. It must be noted that the smoothness of solutions depends essentially on the nature of constraints (the set ${\mathcal K}$ ), and establishing the optimal smoothness requires usually great effort.

One of important applied problems described by a variational inequality is the Signorini problem of deformation under external forces of an elastic body resting on a rigid surface. In the simplest scalar setup this is the problem of the minimization of the functional (1) on the set⁷

$\begin{equation*} {\mathcal K}=\{u\in W^1_m(\Omega)\mid u\geqslant \psi \text{ on }\Gamma \ \text{and}\ u=g \text{ on } \partial\Omega\setminus\Gamma\}. \end{equation*} \notag$

Here

$m\geqslant 1$ ,

$\Gamma\subset\partial\Omega$ is an open set,

$\psi,g\in W^1_m(\Omega)$ are given functions,

$\psi\leqslant g$ on

$\Gamma$ , and the integrand

$F(x,u,p)$ is a function convex in

$p$ such that

$\begin{equation*} F(x,u,p)\sim |p|^m \quad \text{as}\ \ |p|\to \infty. \end{equation*} \notag$

It is known that for smooth data the optimal smoothness of the solution of such a problem is belonging of its first derivatives to the space

${\mathcal C}^{1/2}(\overline{\Omega})$ .

The Signorini problem for elliptic and parabolic equations and diagonal systems of equations was considered in [21]–[27]. In particular, in [22] Hölder estimates for the gradients of solutions were obtained under weaker assumptions than in papers by L. Caffarelli (1979) and D. Kinderlehrer (1981). The minimal assumptions about the obstacle $\psi$ in that paper enabled R. Schumann (1989) to obtain the analogous result in the vector case.

Uraltseva and Arkhipova investigated the regularity (up to the optimal regularity) of the solution of a scalar Signorini problem with two obstacles (an upper and a lower one) for various classes of operators. The case when these obstacles coincide on a part of the boundary was not excluded. They also obtained results on the regularity of solutions of the Signorini problem for diagonal system in the case when the ‘obstacle’ on the boundary of the domain can be described in terms of the belonging of the solution ${\mathbf u}=(u^1,\dots,u^N)$ , $N>1$ , to a convex set $\mathbb{K}\subset \mathbb{R}^N$ . A partial survey of results on the regularity of solutions to variational inequalities was presented in [28].

5. Free boundary problems

Since the late 1990s, Uraltseva’s primary interests have been concentrated on regularity in free boundary problems.

The simplest problems of this type are related to variational inequalities. For instance, in the model obstacle problem on the minimization of the functional

$\begin{equation*} J(u):=\int_\Omega \biggl(\frac{|Du|^2}{2}+u\biggr)\,dx \end{equation*} \notag$

over the set

$\begin{equation*} {\mathcal K}=\{u\in W^1_2(\Omega)\mid u\geqslant 0 \text{ in } \Omega, \ u=g \text{ on } \partial\Omega\}, \qquad g\geqslant0, \end{equation*} \notag$

the necessary condition of a minimum leads to the equation

$\begin{equation} \Delta u=\chi_{\{u>0\}}\quad\text{in } \Omega. \end{equation} \tag{7}$

Thus, the domain

$\Omega$ is partitioned into two sets not known in advance: on one of them the solution is positive and satisfies the equation

$\Delta u=1$ , while on its complement

$u=|Du|=0$ .

The interface $\Gamma$ between these sets is called a free boundary. If the function $g$ in the boundary condition vanishes on a subset of $\partial\Omega$ , then $\Gamma$ can have common points with $\partial\Omega$ . The analysis of the behaviour of both solutions and free boundaries in a neighbourhood of touching points is quite a difficult problem. In the papers [29]–[31] (the first of which was joint with Uraltseva’s former student D. E. Apushkinskaya) Uraltseva proved that if $\partial\Omega$ is smooth in a neighbourhood of a touching point, then $\Gamma\in{\mathcal C}^1$ , and the angle between $\Gamma$ and $\partial\Omega$ is zero (that is, they share the tangent plane). The starting idea was perhaps inspired by the joint works with Oliker described above. Note also that $\Gamma\in{\mathcal C}^1$ is the optimal smoothness in this problem: an example in [30] shows that, in general, the first derivatives of the function defining $\Gamma$ fail the Dini condition.

In [32] and [33] (the latter paper was joint with H. Shahgholian) Uraltseva studied free boundary problems without the assumption that the solution is non-negative (such problems arise, for instance, in the problem of harmonic extension in potential theory). In this case, in place of (7) one considers the equation⁸

$\begin{equation} \Delta u=\chi_{\Omega(u)}\quad\text{in}\ \Omega \end{equation} \tag{8}$

with Dirichlet boundary condition, where

$\Omega(u)$ is the complement to the set where

$u=|Du|=0$ .

This equation looks quite similar to (7), but in fact, as we do not assume that $u$ is positive on $\Omega(u)$ , it is much more difficult to investigate. Nonetheless, a similar result was proved in [33]: the free boundary is ${\mathcal C}^1$ -smooth in a neighbourhood of points where it touches $\partial\Omega$ .

In a series of papers, of which we only mention [34] and [35], Uraltseva and her coauthors investigated similar questions for parabolic problems. It turns out that outside a neighbourhood of the touching points the free boundary is a ${\mathcal C}^{1,\alpha}$ -surface for some $\alpha\in(0,1)$ , while at touching points it can occur to be just strictly Lipschitz continuous (at least with respect to $t$ ). To deduce these results the authors obtained, in particular, new (local) versions of so-called monotonicity formulae, which are of importance in this circle of topics.

Another (two-phase) free boundary problem arises in minimizing the functional

$\begin{equation*} J(u):=\int_\Omega\biggl(\frac{|Du|^2}{2}+\lambda_+ u^+ + \lambda_- u^-\biggr)\,dx \end{equation*} \notag$

(here

$\lambda_{\pm}>0$ are Lipschitz functions in

$\Omega$ and

$u^{\pm}=\max\{\pm u,0\}$ ). In this case a necessary minimum condition is the equation

$\begin{equation} \Delta u =\lambda_+\chi_{\{u>0\}}-\lambda_-\chi_{\{u<0\}} \quad\text{in}\ \Omega \end{equation} \tag{9}$

with an appropriate Dirichlet condition.

Among Uraltseva’s (joint) works on this problem and its parabolic analogue, we point out [36] and [37], where the behaviour of the parts $\partial\{u>0\}$ and $\partial\{u<0\}$ of the free boundary in a neighbourhood of a point of their intersection (which we set to be the origin for convenience) was considered. It is obvious from the implicit function theorem that if $u(0)=0$ and $Du(0)\ne0$ , then $\partial\{u>0\}=\partial\{u<0\}$ is a smooth surface in some neighbourhood. On the other hand, if $Du(0)=0$ , then the origin can be a branch point of the free boundary.

In the elliptic two-phase problem the authors proved that in a neighbourhood of a branch point both boundaries $\partial\{u>0\}$ and $\partial\{u<0\}$ are ${\mathcal C}^1$ -surfaces touching one the other at this point. Moreover, the first derivatives of the functions defining these surfaces fail the Dini condition in general. In the parabolic problem these surfaces are strictly Lipschitz in the direction of the $t$ -axis and ${\mathcal C}^1$ -smooth with respect to the space variables.

These results concern the behaviour of solutions and the free boundary inside the domain. For points on the boundary (or, in the non-stationary case, on the parabolic boundary of the cylinder), in a series of Uraltseva’s papers (some of which were written with Apushkinskaya) the optimal smoothness of solutions (which is $W^2_\infty(\Omega)$ and $W^{2,1}_\infty(\Omega\times(0,T))$ , respectively) was found. Of these papers we mention [38] and [39]. In this case the question of the smoothness of the free boundary is still open.

In [40] the authors considered the vectorial free boundary problem of the minimization of the functional

$\begin{equation*} J({\mathbf u}):= \int_{\Omega}(|\nabla {\mathbf u}|^2+2|{\mathbf u}|)\,dx,\qquad {\mathbf u}\in W^1_2(\Omega\to \mathbb R^N), \quad {\mathbf u}\big|_{\partial\Omega}={\mathbf g}. \end{equation*} \notag$

Then the necessary condition of a minimum is expressed by the system

$\begin{equation*} \Delta u_i=\frac{u_i}{|{\mathbf u}|}\quad\text{in}\ \Omega, \qquad i=1,\dots, N. \end{equation*} \notag$

It is easy to see that (9) can be regarded as a special case of this system by setting

$\begin{equation*} \lambda_{\pm}\equiv1,\quad u_1=u^+,\quad\text{and}\quad u_2=u^-. \end{equation*} \notag$

The authors of [40] showed that under a natural geometric assumption the free surface in this problem is

${\mathcal C}^{1,\alpha}$ -smooth for some

$\alpha\in(0,1)$ .

The techniques developed by Uraltseva for two-phase problems found applications in the investigations of models with spatially distributed hysteresis. In joint papers with Apushkinskaya [41]–[43], for the corresponding problem they established an optimal smoothness of solutions and described the structure of the free boundary (which can be more complicated here than in the preceding problems); they also deduced a monotonicity formula.

Part of Uraltseva’s results on free boundary problems were included in the monograph [44] (joint with A. Petrosyan and H. Shahgholian), highly appreciated by experts.

Uraltseva’s research achievements have repeatedly been awarded, both in our country and overseas. For their results on Hilbert’s 19th and 20th problems Ladyzhenskaya and Uraltseva received the P. L. Chebyshev Prize of the Academy of Sciences of the USSR (1966, for their monograph [5]) and the State Prize of the USSR (1969). For the cycle of papers on free boundary problems Uralyseva was awarded the P. L. Chebyshev Prize of the Government of St Petersburg (2017). She was twice an invited speaker at International Congresses of Mathematicians (in 1970 and 1986), and in 2005 she was chosen as the Lecturer of the European Mathematical Society. She is a Honoured Scientist of the Russian Federation, a Honoured Professor of St Petersburg University, a Honorary Doctor of the KTH Royal Institute of Technology in Stockholm, and a winner of the Humboldt Research Award.

During her work in Leningrad/St Petersburg University Uraltseva delivered quite a number of general and special courses of lectures, which also covered her own original results, including the most recent ones. She was an advisor for 13 Ph.D.’s, four of which defended subsequently their D.Sc. theses. In collaboration with Solonnikov, she wrote a textbook on embedding theorems for Sobolev spaces [45]. Uraltseva’s research and teaching activities made her a recognized leader of the St Petersburg school in nonlinear partial differential equations, the school of Ladyzhenskaya and Uraltseva. A partial survey of the results obtained in this school for the last 50 years was published in [46].

For many years Uraltseva was a member of the board of the St Petersburg mathematical society, of which she was a vice-president in 1998–2008. In 2024 she was elected a honourary member of the Society.

For more than ten years Uraltseva supervised the world known St Petersburg City Seminar on mathematical physics, named after V. I. Smirnov [47]. We should also mention her work as the chairman of the panel on partial differential equations of the International Congress of Mathematicians (1998), the President of the Prize Committee of the European Congress of Mathematics (2004), an expert of the European Research Council, the Russian Foundation for Basic Research, and the Russian Science Foundation, the executive editor of the collections of articles Trudy Sankt-Peterburgskogo Matematicheskogo Obshchestva and Problemy Matematicheskogo Analiza, a member of the editorial boards of the journals Algebra i Analiz,⁹ Vestnik Sankt-Peterburgskogo Gosudarstvennogo Universiteta, and Lithuanian Mathematical Journal, a member of programm committees of many international conferences.

During her life in mathematics Nina Uraltseva made many friends all around the world. Conferences in her honour were organized in Russia, Sweden, and Portugal. The international online conference “Friends in PDEs” dedicated to her 90th birthday took place in May 2024. Its participants observed that Uraltseva’s personal charisma and kindness were felt by every person who had ever occurred within her field of attraction.

On behalf of her students and colleagues, we congratulate Nina Nikolaevna Uraltseva on her 90th birthday and wish her good health and new interesting problems to solve.



Bibliography

1.	D. Apushkinskaya, A. Petrosyan, and H. Shahgholian, “Nina Nikolaevna Uraltseva”, Notices Amer. Math. Soc., 69:3 (2022), 385–395
2.	N. N. Ural'tseva, “Regularity of solutions of multidimensional elliptic equations and variational problems”, Soviet Math. Dokl., 1 (1960), 161–164
3.	N. N. Ural'tseva, “Boundary value problems for quasilinear elliptic equations and systems with divergent principal part”, Soviet Math. Dokl., 3 (1963), 1615–1618
4.	N. N. Ural'tseva, “General quasilinear second-order equations and certain classes of systems of elliptic equations”, Soviet Math. Dokl., 3 (1963), 1399–1402
5.	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968, xviii+495 pp. ; French transl. O. A. Ladyzhenskaya and N. N. Ural'tseva, Equations aux dérivées partielles de type elliptique, Monographies Universitaires de Mathématiques, 31, Dunod, Paris, 1968, xix+450 pp.
6.	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, 2nd revised ed., Nauka, Moscow, 1973, 576 pp. (Russian)
7.	O. A. Ladyženskaya, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968, xi+648 pp.
8.	A. I. Nazarov and N. N. Ural'tseva, “The Harnack inequality and related properties for solutions of elliptic and parabolic equations with divergence-free lower-order coefficients”, St. Petersburg Math. J., 23:1 (2012), 93–115
9.	O. A. Ladyzhenskaya and N. N. Ural'tseva, “Estimate of the Hölder norm of the solutions of second-order quasilinear elliptic equations of the general form”, J. Soviet Math., 21:5 (1983), 762–768
10.	O. A. Ladyzhenskaya and N. N. Ural'tseva, “Estimates of $\max\|u_x\|$ for the solutions of quasilinear elliptic and parabolic equations of the general form and existence theorems”, J. Soviet Math., 32 (1986), 486–499
11.	O. A. Ladyzhenskaya and N. N. Ural'tseva, “A survey of results on the solubility of boundary-value problems for second-order uniformly elliptic and parabolic quasi-linear equations having unbounded singularities”, Russian Math. Surveys, 41:5 (1986), 1–31
12.	N. N. Ural'tseva, “Estimates of derivatives of solutions of elliptic and parabolic inequalities”, Proceedings of the international congress of mathematicians (Berkeley, CA 1986), v. 1, 2, Amer. Math. Soc., Providence, RI, 1987, 1143–1149
13.	N. N. Ural'tseva, “Degenerate quasilinear elliptic systems”, Semin. Math., 7, V. A. Steklov Math. Inst., Leningrad, 1968, 83–99
14.	N. N. Ural'tseva and A. B. Urdaletova, “The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations”, Vestn. Leningr. Univ., Math., 16 (1984), 263–270
15.	O. A. Ladyzhenskaya and N. N. Ural'tseva, “Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations”, Comm. Pure Appl. Math., 23:4 (1970), 677–703
16.	N. N. Uraltseva, “On the non-uniformly quasilinear elliptic equations”, Actes du congrès international des mathématiciens (Nice 1970), v. 2, Gauthier-Villars Éditeur, Paris, 1971, 885–891
17.	O. A. Ladyzhenskaya and N. N. Uraltseva, “Local gradient estimates for solutions of a simplest regularization of a class of nonuniformly elliptic equations”, J. Math. Sci. (N. Y.), 84:1 (1997), 862–872
18.	N. N. Ural'tseva, “Solvability of the capillary problem”, Vestn. Leningr. Univ., Math., 6 (1979), 363–375
19.	N. N. Ural'tseva, “Solvability of the capillary problem. II”, Vestn. Leningr. Univ., Math., 8 (1980), 151–158
20.	V. I. Oliker and N. N. Ural'tseva, “Long time behavior of flows moving by mean curvature. II”, Topol. Methods Nonlinear Anal., 9:1 (1997), 17–28
21.	N. N. Ural'tseva, “Strong solutions of the generalized Signorini problem”, Siberian Math. J., 19:5 (1978), 850–856
22.	N. N. Ural'tseva, “Hölder continuity of the gradients of solutions of parabolic equations under boundary conditions of Signorini type”, Soviet Math. Dokl., 31 (1985), 135–138
23.	A. A. Arkhipova and N. N. Ural'tseva, “Regularity of the solutions of diagonal elliptic systems under convex constraints on the boundary of the domain”, J. Soviet Math., 40:5 (1988), 591–598
24.	A. A. Arkhipova and N. N. Ural'tseva, “Limit smoothness of the solutions of variational inequalities under convex constraints on the boundary of the domain”, J. Soviet Math., 49:5 (1990), 1121–1128
25.	A. A. Arkhipova and N. N. Ural'tseva, “Regularity of the solution of a problem with a two-sided constraint on the boundary for elliptic and parabolic equations”, Proc. Steklov Inst. Math., 179 (1989), 1–19
26.	A. A. Arkhipova and N. N. Uraltseva, “Existence of smooth solutions of problems for parabolic systems with convex constraints on the boundary of the domain”, J. Soviet Math., 56:2 (1991), 2281–2285
27.	A. Arkhipova and N. Uraltseva, “Sharp estimates for solutions of a parabolic Signorini problem”, Math. Nachr., 177 (1996), 11–29
28.	N. N. Ural'tseva, “Regularity of solutions of variational inequalities”, Russian Math. Surveys, 42:6 (1987), 191–219
29.	D. E. Apushkinskaya and N. N. Uraltseva, “On the behavior of free boundaries near the boundary of the domain”, J. Math. Sci. (N. Y.), 87:2 (1997), 3267–3276
30.	N. N. Ural'tseva, “ $C^1$ regularity of the boundary of the noncoincident set in the obstacle problem”, St. Petersburg Math. J., 8:2 (1997), 341–353
31.	N. N. Ural'tseva, “On some properties of the free boundary in a neighborhood of the points of contact with a given boundary”, J. Math. Sci. (N. Y.), 101:5 (2000), 3570–3576
32.	N. N. Ural'tseva, “Contact of a free boundary with a fixed boundary”, Sb. Math., 191:2 (2000), 307–315
33.	H. Shahgholian and N. Uraltseva, “Regularity properties of a free boundary near contact points with the fixed boundary”, Duke Math. J., 116:1 (2003), 1–34
34.	D. E. Apushkinskaya, H. Shahgholian, and N. N. Ural'tseva, “Boundary estimates for solutions of the parabolic free boundary problem”, J. Math. Sci. (N. Y.), 115:6 (2003), 2720–2730
35.	D. E. Apushkinskaya, N. Matevosyan, and N. N. Uraltseva, “The behavior of the free boundary close to a fixed boundary in a parabolic problem”, Indiana Univ. Math. J., 58:2 (2009), 583–604
36.	H. Shahgholian, N. Uraltseva, and G. S. Weiss, “The two-phase membrane problem – regularity of the free boundaries in higher dimensions”, Int. Math. Res. Not. IMRN, 2007:8 (2007), rnm026, 16 pp.
37.	H. Shahgholian, N. Uraltseva, and G. S. Weiss, “A parabolic two-phase obstacle-like equation”, Adv. Math., 221:3 (2009), 861–881
38.	N. N. Uraltseva, “Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities”, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, 220, Adv. Math. Sci., 59, Amer. Math. Soc., Providence, RI, 2007, 235–246
39.	D. E. Apushkinskaya and N. N. Uraltseva, “Uniform estimates near the initial state for solutions of the two-phase parabolic problem”, St. Petersburg Math. J., 25:2 (2014), 195–203
40.	J. Andersson, H. Shahgholian, N. N. Uraltseva, and G. S. Weiss, “Equilibrium points of a singular cooperative system with free boundary”, Adv. Math., 280 (2015), 743–771
41.	D. E. Apushkinskaya and N. N. Uraltseva, “On regularity properties of solutions to the hysteresis-type problem”, Interfaces Free Bound., 17:1 (2015), 93–115
42.	D. E. Apushkinskaya and N. N. Uraltseva, “Free boundaries in problems with hysteresis”, Philos. Trans. Roy. Soc. A, 373:2050 (2015), 20140271, 10 pp.
43.	D. E. Apushkinskaya and N. N. Uraltseva, “Monotonicity formula for a problem with hysteresis”, Dokl. Math., 97:1 (2018), 49–51
44.	A. Petrosyan, H. Shahgholian, and N. Uraltseva, Regularity of free boundaries in obstacle-type problems, Grad. Stud. Math., 136, Amer. Math. Soc., Providence, RI, 2012, x+221 pp.
45.	V. A. Solonnikov and N. N. Uraltseva, “Sobolev spaces”, Selected chapters in analysis and higher algebra, Publihing house of Leningrad State University, Leningrad, 1981, 129–199 (Russian)
46.	D. E. Apushkinskaya, A. A. Arkhipova, A. I. Nazarov, V. G. Osmolovskii, and N. N. Uraltseva, “A survey of results of St. Petersburg State University research school on nonlinear partial differential equations. I”, Vestnik St. Petersburg Univ. Math., 57:1 (2024), 1–22
47.	D. E. Apushkinskaya and A. I. Nazarov, “V. I. Smirnov Seminar is 75!”, Boundary value problems of mathematical physics and related questions in function theory. 50, Zap. Nauchn. Semin. Sankt-Peterburg. Otdel. Mat. Inst. Steklov, 519, St Petersburg Department of the Steklov mathematical Institute, St Petersburg, 2022, 5–9 (Russian)

Citation: D. E. Apushkinskaya, A. A. Arkhipova, V. M. Babich, G. S. Weiss, I. A. Ibragimov, S. V. Kislyakov, N. V. Krylov, A. A. Laptev, A. I. Nazarov, G. A. Seregin, T. A. Suslina, H. Shahgholian, “On the 90th birthday of Nina Nikolaevna Uraltseva”, Russian Math. Surveys, 79:6 (2024), 1119–1131

Citation in format AMSBIB

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\paper On the 90th birthday of Nina Nikolaevna Uraltseva

\jour Russian Math. Surveys

\yr 2024

\vol 79

\issue 6

\pages 1119--1131

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1. Equations in divergence form and calculus of variations

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3. Degenerate equations

4. Variational inequalities

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