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 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 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 particular, 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. The typical conditions reads B (x0) ⋐ Ω M with the ball , is a cuto function, and 2 is a bound for max ð∇uð. ða(x, u, p)ð ≤ ðpð + b(x)ðpð + Φ(x), Using similar ideas, Uraltseva was able to deduce where is a constant and b, Φ ∈ Lq(Ω), q > n. Uralt- the existence and regularity of solutions for the class seva and collaborators were able to establish the exis- of quasilinear uniformly elliptic equations in divergence tence and up to the boundary C1, regularity of W2,n form strong solutions of the problem, vanishing on )Ω (pro- ) (a (x, u, ∇u)) + a(x, u, ∇u) = 0 xi i (2) vided the latter is suciently regular). The proofs were based on the extension of methods of Ladyzhenskaya under natural growth conditions on ai(x, u, p), a and some of their partial derivatives, which were mainly and Uraltseva already in their books [18, 30], as well needed for proving the bounds on max ð∇uð. These re- as those of Krylov and Safonov using the Aleksandrov- sults were further rened in the joint works with Olga Bakelman-Pucci (ABP) estimate, in the elliptic case, and Ladyzhenskaya [19–22,26] as well as in [46], for the case a parabolic version of the ABP estimate due to Nazarov of Neumann type boundary conditions. The latter pa- and Uraltseva [32], in the parabolic case. per also contained similar results for certain quasilinear Most recent results of Nina Uraltseva in this direction diagonal systems (important, e.g., for the applications are in the joint work with Alexander Nazarov [33] on in harmonic maps). the linear equations in divergence form Quasilinear uniformly elliptic equations in nondiver- )xi (aij(x)uxj ) + bi(x)uxi = 0 in Ω gence form and their parabolic counterparts. Their goal was to

aij(x, u, ∇u) uxixj + a(x, u, ∇u) = 0 (3) nd conditions on the lower order coecients b =

3 or, equivalently, are the minimizers of the energy functional Ê ð∇uðpdx. Ω The diculty here lies in the fact that the p-Laplace equation (5) degenerates at the points where the gra- dient vanishes and that the solutions are not generally twice dierentiable in the Sobolev sense. As indicated in her paper, this problem was posed to Nina Uraltseva by Yurii Reshetnyak in relation with the study of quasi- conformal mappings in higher dimensions. Uraltseva has obtained the C1, regularity of p- harmonic functions as an application of the Hölder reg- Figure 3: The famous books: the iconic green Russian ularity of the solutions of the degenerate quasilinear editions of the Elliptic (2nd ed., 1973) and Parabolic system ) (a (x, u) u ) = 0 (1967) versions of Uraltseva’s books with Ladyzhenskaya xi ij xj and Solonnikov with scalar coecients aij satisfying the degenerate ellipticity condition (b1, … , bn) that guarantee the validity of classical re- (ðuð)ðð2 ≤ a (x, u) ≤ (ðuð)ðð2, sults such as the strong maximum principle, Harnack’s ij i j inequality, and Liouville’s theorem. It was shown by q with ≥ 1 and a nonnegative increasing function () Trudinger [43] that such results hold when b ∈ L , s satisfying () ≤ () for ≥ 1 and s > 0. q > n. Motivated by applications in uid dynamics, in Unfortunately, despite the utmost importance of this one of their theorems, Nazarov and Uraltseva showed result, Nina Uraltseva’s proof remained unknown out- that under the additional assumption side of Soviet Union. In 1977, 9 years later, it was in- dependently reproved by Karen Uhlenbeck [44]. Other div b = 0, (4) proofs were given by Craig Evans [14], John Lewis [31], 1 < p ≤ 2 the condition on b can be relaxed to being in the Morrey who extended the range of exponents to , Di space Benedetto [13] and Tolksdorf [42], who both extended it to the case of general degenerate quasilinear equations sup rq−n Ê ðbðq < ∞ in divergence form. 0 0 Br(x )⊂Ω Br(x ) for some n∕2 < q ≤ n. In the borderline case q = 2.2 Geometric Equations n, the Morrey space above is locally the same as Ln. In paper [27], Ladyzhenskaya and Uraltseva developed Remarkably, in that case the divergence free condition a method of local a priori estimates for nonuniformly (4) on b can be dropped when n ≥ 3, i.e., b ∈ Ln alone loc elliptic and parabolic equations, including the equations is sucient to have the classical theorems; moreover, of minimal surface type this result is optimal. In dimension n = 2, to drop (4) 1∕2 one needs a stronger condition b ln (1 + ðbð) ∈ L2 . ∇u loc div √ = a(x, u, ∇u) in Ω. 1 + ð∇uð2 2 Nonuniformly Elliptic and Parabolic A particular case with a(x, u, ∇u) = u√, > 0, together Equations 2 with the Neumann type condition )u∕ 1 + ð∇uð = z 2.1 Degenerate Equations on )Ω, ðzð < 1, is known as the capillarity problem. The boundary estimates, as well as the existence of classical Nina Uraltseva has also made a pioneering work on the solutions for such problems were obtained in [49, 51, 52, regularity theory for degenerate quasilinear equations. 54]. It is remarkable that the estimates in the last two A particular result in this direction is her 1968 proof [48] papers required only the smoothness of the domain Ω, of the C1, -regularity of p-harmonic functions, p > 2, but not its convexity. which are the weak solutions of the p-Laplace equation In 1990s, in a series of joint works [34–37] with Vladimir Oliker, Nina Uraltseva has studied the evo- div(ð∇uðp−2∇u) = 0 in Ω, (5) lution of surfaces S(t) given as graphs u = u(x, t) over a

4 bounded domain Ω ⊂ ℝn with the speed depending on of the C1, regularity for the solution of the Signorini the mean curvature of the S(t) under the condition that problem in the vectorial case. the boundary of the surface S(t) is xed. More precisely, Below, we give a more detailed description of some of they have considered a parabolic PDE of the type her most impactful results in this direction. √ 2 ∇u ut = 1 + ð∇uð div √ in Ω × (0, ∞) 3.1 Problems with unilateral constraints 1 + ð∇uð2 Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain with a smooth with the boundary condition u(x, t) = (x) on )Ω × boundary and S a relatively open nonempty subset of )Ω. 1,2 (0, ∞) and initial condition u(x, 0) = u0(x). Even in Suppose we are also given two functions , g ∈ W (Ω) the stationary case, when this problem is the Dirich- satisfying g ≥ on S (in the sense of traces). Consider let problem for the mean curvature equation, the exis- then a closed convex subset K ⊂ W1,2(Ω) dened by tence of up to the boundary classical solutions requires 1,2 a geometric condition on the domain Ω, namely, the K ≔ {v ∈ W (Ω) ∶ v ≥ on S, v = g on )Ω ⧵ S}. nonnegativity of the mean curvature of )Ω. For such K domains, Huisken [15] has shown the existence of the In other words, consists of functions that need to stay S classical solutions of the evolution problem and proved above , called a boundary (or thin) obstacle, on and g )Ω ⧵ S u ∈ K that the surfaces S(t) converge to a classical minimal equal to on . Then, one wants to nd surface S as t → ∞. Oliker and Uraltseva have studied that minimizes the generalized Dirichlet energy this problem without any geometric conditions on the

domain Ω. For this purpose, they introduced a notion J(v) = Ê aij(x)vxj vxi + 2f(x)u, of a generalized solution to the parabolic problem (as Ω

a limit of regularized problems). They have proved its where aij(x) are uniformly elliptic coecients and f is a u(⋅, t) → Φ t → ∞ existence and convergence as to a certain function. Equivalently, the minimizer u satises Φ generalized solution of the stationary problem, in the the variational inequality sense that Φ minimizes the area functional √ u ∈ K, Ê a (x)u (v − u) Ê 1 + ð∇uð2 + Ê ðu − ð ij xj xi Ω )Ω Ω + f(x)(v − u) ≥ 0, for any v ∈ K. among all competitors in W1,1(Ω). Such minimizer Φ is unique, but may dier from the Dirichlet data on the In turn, it is equivalent to the following boundary value “bad” part of the boundary where the mean curvature problem is negative. The study of the behavior of the minimizer ) (a (x)u ) = f(x) Ω, near the “contact points” on the boundary where Φð)Ω xi ij xj in “detaches” from later served as one of Uraltseva’s mo- u = g on )Ω ⧵ S, tivations for studying the touch between free and xed u ≥ , )Au ≥ 0, (u − ))Au = 0 on S, boundaries, see Sec 4.1. to be understood in the appropriate weak sense, where A 3 Variational Inequalities ) u ≔ aij(x)juxj is the conormal derivative of u on )Ω, with = (1, … , n) being the outward unit normal. Another area in which Nina Uraltseva has made signi- The conditions on S are known as the Signorini com- cant contributions are variational inequalities, including plementarity conditions and are remarkable by the fact variational problems with convex constraints that often that they imply that exhibit a priori unknown sets known as free boundaries. A An important example is the Signorini problem from either u = or ) u = 0 on S, elasticity which describes equilibrium congurations of an elastic body resting on a rigid frictionless surface. yet the exact sets where the rst or the second equality In a series of papers [7–11,50,53,55,56], partially with holds are unknown. The interface Γ between these sets Arina Arkhipova, Nina Uraltseva studied elliptic and in S is called free boundary (see Fig. 4). The study of parabolic variational inequalities with unilateral and the free boundary is one of the main objectives in such bilateral boundary constraints, known as the boundary problems (see Sec. 4 for Uraltseva’s contributions in that obstacle problems which can be viewed as scalar ver- direction), yet the regularity of the solutions u is a chal- sions of the Signorini problem. Ultimately, these results lenging problem by itself and is often an important step played a fundamental role in Schumann’s proof [39] towards the study of the free boundary.

5 observes that as a consequence of the Signorini comple- S mentarity conditions, one has

u = uxi uxn = 0 on {xn = 0} ∩ B(x0), )Au ≥ 0 for all i = 1, … , n − 1 and hence either the normal Ω v = u v = u Γ derivative xn or all tangential derivatives xi , 0 i = 1, … , n−1, vanish at least on half of {xn = 0}∩B(x ) u > (by measure). This allows to apply Poincare’s inequality A ) u = 0 in one of the steps and obtain a geometric improvement of the Dirichlet energy for v going from radius to ∕2. By iteration, this gives that either

nÉ−1 Ê ð∇u ð2 ≤ Cn−2+2 , Figure 4: Boundary obstacle problem xi or (6) 0 i=1 Ω∩B(x )

Ê ð∇u ð2 ≤ Cn−2+2 . One of the theorems of Nina Uraltseva [55, 56] states xn (7) Ω∩B (x0) that when holds, with C depending on the distance from x0 to a ∈ W1,q(Ω), ij )Ω ⧵ S. However, using the PDE satised by u, it is easy 1,2 2,q ∈ W (Ω) ∩ Wloc (Ω ∪ S), to see that (6) implies (7), and hence (7) always holds. C1, u f ∈ Lq(Ω), From there, the -regularity of follows by standard results for the solutions of the Neumann problem. for some q > n, then 3.2 Diagonal systems u ∈ C1, (Ω ∪ S), loc The results described above were extended, in joint with a universal exponent ∈ (0, 1). Prior to this result, works with Arina Arkhipova [9,11], to the problem with ≤ S similar conclusion was known only under higher regu- two obstacles − + on , that corresponds to the larity assumptions on the coecients and the obstacle in constraint set the works of Caarelli [12] and Kinderlehrer [16]. The K = {v ∈ W1,2(Ω) ∶ ≤ u ≤ S, lower regularity assumptions in Uraltseva’s result, par- − + on ticularly on the obstacle , were instrumental in Schu- u = g on )Ω ⧵ S}. mann’s proof of the corresponding result in the vectorial case [39]. The parabolic counterpart of Uraltseva’s While substantial diculties arise near the set where theorem, with similar assumptions on the coecients − = +, the results are as strong as in the case of a and the obstacle was established later in a joint work of single obstacle. In their further work, Arkhipova and Arkhipova and Uraltseva [7]. Uraltseva [8,10] studied related problems for quasilinear The idea of Uraltseva’s proof is based on an interplay elliptic systems with diagonal principal part. To describe V = W1,2(Ω; ℝN) ∩ L∞(Ω; ℝN) between De Giorgi type energy inequalities and the Sig- their results, let and norini complementarity condition. Locally, near x0 ∈ S, K = {u ∈ V ∶ u(x) ∈ K(x) for every x ∈ )Ω}, one can assume that S = {xn = 0} and = 0. First, working with the regularized problem, one can estab- where K(x) are given convex subset of ℝN for every v = ±u i = 1, … , n lish that for any partial derivative xi , , x ∈ )Ω. Then consider the variational inequality of the there holds an energy inequality (similar to (1) in the type unconstrained case) u ∈ K, Ê aij(x, u)ux + bi(x, u) (v − u)x 2 2 2 2 j i Ê ð∇vð ≤ C Ê (v − k) ð∇ð Ω Ak, Ak, + f (x, u, ∇u)(v − u) ≥ 0, 1−2∕q + C0ðAk,ð , for any v ∈ K,

for any k > 0, 0 < < 0, and a cuto function in where aij are scalar uniformly elliptic coecients, bi 0 0 B(x ), where Ak, = {v > k} ∩ B(x ) ∩ Ω. Next, one and f are N-dimensional vector functions and f (x, u, p)

6 + grows at most quadratically in p. We note that the prob- B1 lem with two obstacles − ≤ + on )Ω ts into this Ω(u) framework with N = 1 and K(x) = [ −(x), +(x)]. As- sume now that the convex sets K(x) are of the form ∆u = 1 ΓΓ K(x) = T(x)K + g(x), 0 u = 0 u = 0 ð∇uð = 0 ð∇uð = 0 N where K0 is a convex set in ℝ with a nonempty interior and a smooth (C2) boundary, T(x) is an orthogonal N×N Π u = 0 matrix, and g(x) is an N-dimensional vector. A theorem Figure 5: Touch between the free boundary Γ = )Ω(u) of Arkhipova and Uraltseva [10] then states that when 2,q and the xed boundary Π in problem (8) the entries of T and g are extended to W functions in 1,q Ω, q > n, aij(⋅, u) and bi(⋅, u) are in W (Ω), uniformly in u and have at most linear growth in u, and f has at the free boundary touches the xed boundary tangen- most quadratic growth in p, then tially. The idea seemed to be inspired by related works with Oliker (see Sec. 2.2) and the Dam-problem in ltra- u ∈ C1, (Ω ∪ S), loc tion. During the Potential theory program at Institute u Ω ∪ S provided is Hölder continuous in . The Hölder Mittag-Le er (1999-2000) she started working on free u continuity assumption on can be replaced by a bound boundary problems that originated in potential theory. Ω on the oscillation in and a local uniqueness of the Specically the harmonic continuation problem in po- solutions, which is also necessary for the continuity of tential theory, that was strongly tied to obstacle problem, the solutions of the nonlinear systems of the type but with the lack of having a sign for the solution function. The simplest way to formulate this problem is as ) (a (x, u)u + b (x, u)) xi ij xj i follows: + f (x, u, ∇u) = 0 in Ω. + ∆u = Ω(u) in B1 , c For a more complete overview of Uraltseva’s results on with Ω(u) ≔ {u = ð∇uð = 0} (8) variational inequalities, we refer to her own survey paper u = 0 on Π ∩ B1, [58]. + where B1 = {ðxð < 1, x1 > 0} and Π = {x1 = 0}; see 4 Free Boundary Problems Fig. 5. The question of interest was the behavior of the free boundary Γ = )Ω(u) close to the xed boundary Π. In the last 25 years, Uraltseva’s work has dealt with regu- In [3], and several follow up papers in parabolic regime, she shows that the free boundary Γ is a graph of larity issues arising in free boundary problems. She has 1 + developed powerful techniques, which has led to prov- a C -function close to points on Π, where Γ∩B1 touches ing the optimal regularity results for solutions and for Π, or comes too close to Π. free boundaries. She has systematically studied how the To prove this, and the related parabolic results, there free boundaries approach the xed boundaries [40], and was a need for developing new tools and approaches. has developed tools to study free boundary problems for This was possible partly due to the availability of mono- weakly coupled systems [2], as well as two-phase prob- tonicity formulas such as that of Alt, Caarelli, and Friedman [1]. One version of the latter asserts that for lems [41]. The graduate textbook “Regularity of Free 0 continuous subharmonic functions ℎ1, ℎ2 in BR(x ), sat- Boundaries in Obstacle-Type Problems” [38], written in 0 0 collaboration with two of us, contains these and related isfying ℎ1ℎ2 = 0, and ℎ1(x ) = ℎ2(x ) = 0, we have results. '(r) ↗, for 0 < r < R, where Some of Uraltseva’s major contributions (results, ap- 0 0 '(r) = (r, ℎ1, x ) (r, ℎ2, x ) (9) proaches) in free boundary problems are addressed below in more detail. with 1 ð∇ℎ ð2dx 4.1 Touch between free and xed boundary (r, ℎ , x0) ≔ Ê i . i 4 0 n−2 r B(x0,r) ðx − x ð In [4] (joint with one of us) and her follow up paper [59], Uraltseva studied the obstacle problem close to a Dirich- One can use the monotonicity of the function '(r) to let data, for smooth boundaries, where she proves that prove several important properties for u and the free

7 − boundary. Indeed, one rst extends u to be zero in B1 = {ðxð < 1, x1 < 0} and applies the monotonicity formula + − (9) to ℎ1 = ()eu) and ℎ2 = ()eu) , where e is any vector tangent to the plane {x1 = 0}. Using the fact that at least one of the sets {±)eu > 0} has positive volume u > 0 x0 density at , we shall have x0 u < 0 ∆u = + 0 4 c0ð∇)eu(x )ð = lim '(r) ≤ '(1) ≤ C0. r→0 ∆u = − − u = 0 Combining this with equation (8) we obtain the bound 0 1,1 for ux1x1 (x ). From here, the uniform C regularity for u B+ in 1∕2 follows. The C1,1 regularity is instrumental for any analysis of the properties of the free boundary. Indeed, to study 0 the free boundary at points where it touches the xed Figure 6: Two-phase problem: branch point x boundary, one needs to rescale the solution quadrati- 0 2 cally ur(x) = u(rx + x )∕r , which keeps the equation 6 invariant. Indeed, this scaling and “blow-up” brings 1 Br (0) and ){u < 0} ∩ Br (0) are C -surfaces, that touch one to a global setting of equation (8) in ℝn , where so- 0 0 + each other tangentially at x0. lutions can be classied (in a rotated system) as one of the following: The proof of this and several similar results (also in parabolic setting) relies heavily on the monotonicity x2 1 function ' mentioned above as well as on balanced en- (i) u(x) = + ax1x2 + x1, (a > 0, ∈ ℝ) 2 ergy functional + 2 ((x1 − a) ) (ii) u(x) = , (a > 0). 2 Φ (r) ≔ r−n−2 Ê ð∇uð2 + u+ + u− The proof of the classication of global solutions uses x0 + − Br(x0) an array of geometric tools as well as the monotonicity −n−3 2 function '(r), implying that if {u = 0} ∩ {x1 > 0} ≠ − 2r Ê u , (10) ç, then )eu ≡ 0, for any direction e tangential to Π. )Br(x0) The case when this set is empty is easily handled by Liouville’s theorem. which is strictly monotone in r, unless u is homoge- Once this classication is done, one can argue by indi- neous. The use of these two monotonicity functional in rect methods that the free boundary ){u > 0} ∩ {x1 > 0} combination with geometric tools bring us to the fact approaches the xed one, at touching points, in a tan- that any global solution u0 to the two-phase problem are gential fashion, and that it is a C1-graph locally, which is one-dimensional and, in a rotated and translated system optimal in the sense that in general it cannot be C1,Dini. of coordinates, 4.2 Two-phase obstacle type problems u = + (x+)2 − − (x−)2. If one considers extension of equation (8) into B1, by an 0 2 1 2 1 odd reection, then one obtains a specic example of a general problem that is referred to as two-phase obstacle problem, and is formulated as From here one uses a revised form of the so-called di- rectional monotonicity argument of Luis Caarelli, that ∆u = + {u>0} − − {u<0} in B1(0), in this setting boils down to the fact that close to branch points x0 one can show that in a suitable cone of di- 0 where ± are positive bounded Lipschitz functions. rections C one has )eu ≥ 0, in Br(x ) for e ∈ C and r Fig. 6 illustrates this problem. universal. This in particular implies that the free bound- In [41], Nina Uraltseva (with co-authors) proves that aries ){±u > 0} are Lipschitz graphs locally close to at any branch point x0 ∈ ){u > 0} ∩ ){u < 0} with branch points. u(x0) = ð∇u(x0)ð = 0, the free boundaries ){u > 0} ∩ The approaches here generated further application of 6 Blow-up refers to limr→0 ur(x), whenever it exists. the techniques to problems with hysteresis [5,6].

8 4.3 Free boundaries for weakly coupled systems

In her work with coauthors [2], Uraltseva considers the following vectorial energy minimizing functional E(u) ≔ Ê ð∇uð2 + 2ðuð dx. B1

n Here B1 is the unit ball in ℝ (n ≥ 1), and we mini- 1,2 N mize over the Sobolev space g + W0 (B1; ℝ ) for some smooth boundary values g = (g1, … , gN). The minimizer(s) are vector-valued functions u = (u1, … , uN), with components ui satisfying u ∆u = i , i = 1, … ,N. i ðuð

Since the set {ðuð > 0} competes with the Dirichlet energy, by taking the boundary values small we may obtain {u = 0} ≠ ç, which is in contrast to standard variational problems. The set ){ðuð > 0} is called free boundary. Figure 7: Nina Uraltseva in 2013 One observes that when N = 1 (scalar case) then we fall back to the two-phase problem. ¨ > 0 r > 0 C < ∞ Simple examples of solutions to this problem are coordinates) there exist , 0 , and :

(i) ui = iP(x), with P(x) ≥ 0, ∆P(x) = 1, and ¨ ∑N Ê ður − hð ≤ Cr 2 = 1 i=1 i , )B1(0) 0 for every x ∈ ℛu and every r ≤ r0, + 2 − 2 (ii) ui = i(x1 ) ∕2 + i(x1 ) ∕2, (2-phase) ∑N 2 ∑N 2 where ℛu is the set of free boundary points whose blow- i=1 i = 1, i=1 i = 1, ups are half-spaces. This implies that ℛu is locally in B C1, + 2 1∕2 a -surface. (iii) ui = i(x1 ) ∕2, (1-phase) ∑N 2 = 1 i=1 i . Nina’s Impact Using the vectorial version of the monotonicity formula (10), one can show that u has a quadratic growth away Nina Uraltseva has over 100 publications7 and over 8000 from the free boundary. citations in MathSciNet. Her famous book “Linear and The regularity of the free boundary follows through Quasilinear Equations of Parabolic Type” [17, 18] (joint the homogeneity improvement approach with the so- with Ladyzhenskaya and Solonnikov) has over 4600 ci- called epiperimetric inequality, which is used to show tation, and the elliptic version of this book [25,30] (joint that the functional with Ladyzhenskaya), has over 1600 citation in Math- SciNet. This naturally gives a picture of a mathematician ℳ(v) ≔ Ê (ð∇vð2 + 2ðvð) − Ê ðvð2 with tremendous impact on the led of partial dieren- B1(0) )B1(0) tial equations. Needless to say that, even though there are many new books on the topic of PDEs, these books satises stay equally important and extremely valuable to many Ph.D. students and early career analysts. ðℳ(u ) − ℳ(u )ð ≤ cðr − r ð , > 0, r1 r2 2 1 Nina Uraltseva has, over years, contributed to the

u(x0+rx) mathematical community by serving on many impor- u = x0 u where r r2 , and is such that r is close tant committees; e.g., chairing the PDE Panel of the In- to rotated version of a half-space solution of type h = ternational Congress of Mathematicians in Berlin, Ger- 1 + 2 many, 1998, and the Prize Committee of the European (x1 ) e. 2 Congress of Mathematics in Stockholm, Sweden, 2004. This, in particular, gives uniqueness of the blow-ups, and can be used to show that (in a rotated system of 7See: https://www.scilag.net/profile/nina-uraltseva

9 She also served as an expert for research foundations [6] Darya E. Apushkinskaya and Nina N. Uraltseva, On regularity such as the European Research Council and the Russian properties of solutions to the hysteresis-type problem, Interfaces Free Bound. 17 (2015), no. 1, 93–115. MR 3352792 Foundation for Basic Research. Sharp estimates for solutions of 8 [7] A. Arkhipova and N. Uraltseva, She has been acting as an editor for several journals , a parabolic Signorini problem, Math. Nachr. 177 (1996), 11–29. and has been a frequent visitor of many universities all MR 1374941 0 over the world and presented talks at various interna- [8] A. A. Arkhipova and N. N. Ural tseva, Regularity of solutions of diagonal elliptic systems under convex constraints on the bound- tional conferences, and schools. In her role as a world ary of the domain, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. leading expert in analysis of PDEs she has captured the Inst. Steklov. (LOMI) 152 (1986), no. Kraev. Zadachi Mat. Fiz. i attention of many female students in all areas of mathe- Smezhnye Vopr. Teor. Funktsi˘ı18, 5–17, 181. MR 869237 matics, and attracted them to further pursue research [9] , Regularity of the solution of a problem with a two-sided constraint on the boundary and start a career in mathematics. Her motivational , Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1986), no. vyp. 1, 3–10, 133. MR 841488 talks at many conferences, specially meetings related to [10] , Regularity of the solutions of variational inequalities with “connection to women” has been an important factor for convex constraints on the boundary of the domain for nonlinear attraction of several females to mathematics. operators with a diagonal principal part, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1987), no. vyp. 3, 13–19, 127. MR 928154 The instructional aspect of her work and her dedica- Regularity of the solution of a problem with a two-sided 9 [11] , tion to educating Ph.D. students, as well as unselshly limit on a boundary for elliptic and parabolic equations, Trudy being available to students and colleagues, for discus- Mat. Inst. Steklov. 179 (1988), 5–22, 241, Translated in Proc. sions and brain-storming of their problems, make her Steklov Inst. Math. 1989, no. 2, 1–19, Boundary value problems one of the most prominent and devoted persons to the of mathematical physics, 13 (Russian). MR 964910 [12] L. A. Caarelli, Further regularity for the Signorini problem, mathematical community. Comm. Partial Dierential Equations 4 (1979), no. 9, 1067–1075. Nina Uraltseva has dedicated her life to mathematics, MR 542512 1+ and in her scientic journey through years she has made [13] E. DiBenedetto, C local regularity of weak solutions of degenerate elliptic equations 7 many friends all over the world. Her kind personality, , Nonlinear Anal. (1983), no. 8, 827–850. MR 709038 and utmost politeness on one side and her unbiased [14] Lawrence C. Evans, A new proof of local C1, regularity for solu- style and open mindedness towards diverse mathemati- tions of certain degenerate elliptic p.d.e, J. Dierential Equations cal problems, have made her extremely popular among 45 (1982), no. 3, 356–373. MR 672713 colleagues and students, and not only as a mathemati- [15] Gerhard Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Dierential Equations 77 (1989), no. 2, cian but also as a human being. 369–378. MR 983300 [16] David Kinderlehrer, The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl. (9) 60 (1981), no. 2, 193–212. MR 620584 References 0 [17] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural ceva, ˘ ˘ [1] Hans Wilhelm Alt, Luis A. Caarelli, and Avner Friedman, Vari- Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Izdat. ational problems with two phases and their free boundaries, Trans. “Nauka”, Moscow, 1967. MR 0241821 Amer. Math. Soc. 282 (1984), no. 2, 431–461. MR 732100 [18] , Linear and quasilinear equations of parabolic type, [2] John Andersson, Henrik Shahgholian, Nina N. Uraltseva, and Translations of Mathematical Monographs, Vol. 23, American Georg S. Weiss, Equilibrium points of a singular cooperative sys- Mathematical Society, Providence, R.I., 1968, Translated from tem with free boundary, Adv. Math. 280 (2015), 743–771. MR the Russian by S. Smith. MR 0241822 0 3350233 [19] O. A. Ladyženskaja and N. N. Ural ceva, A boundary-value prob- [3] D. E. Apushkinskaya, H. Shahgholian, and N. N. Uraltseva, lem for linear and quasi-linear parabolic equations, Dokl. Akad. Boundary estimates for solutions of a parabolic free boundary prob- Nauk SSSR 139 (1961), 544–547. MR 0141891 lem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. [20] , Dierential properties of bounded generalized solutions (POMI) 271 (2000), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. of multidimensional quasilinear elliptic equations and varia- Teor. Funkts. 31, 39–55, 313. MR 1810607 tional problems, Dokl. Akad. Nauk SSSR 138 (1961), 29–32. MR 0 [4] D. E. Apushkinskaya and N. N. Ural tseva, On the behavior of 0141874 the free boundary near the boundary of the domain, Zap. Nauchn. [21] , Quasilinear elliptic equations and variational problems Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), in several independent variables, Uspehi Mat. Nauk 16 (1961), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsi˘ı. 26, no. 1 (97), 19–90. MR 0149075 5–19, 253. MR 1359745 [22] , Regularity of generalized solutions of quasilinear ellip- [5] D. E. Apushkinskaya and N. N. Uraltseva, Free boundaries in tic equations, Dokl. Akad. Nauk SSSR 140 (1961), 45–47. MR problems with hysteresis, Philos. Trans. Roy. Soc. A 373 (2015), 0150447 no. 2050, 20140271, 10. MR 3393312 [23] , A boundary-value problem for linear and quasi-linear parabolic equations. I, II, III, Iaz. Akad. Nauk SSSR Ser. Mat. 26 8Editor in Chief for Proceedings of St. Petersburg Math. Society (1962), 5-52; ibid. 26 (1962), 753- 780; ibid. 27 (1962), 161–240. and Journal of Problems in Mathematical Analysis; Member of Edito- MR 0181837 rial Committee for Algebra and Analysis (translated in St. Petersburg [24] , The rst boundary-value problem for quasi-linear second- Mathematical Journal), Vestnik St. Petersburg State University, Lithua- order parabolic equations of general type, Dokl. Akad. Nauk SSSR nian Mathematical Journal 147 (1962), 28–30. MR 0147786 9Uraltseva has supervised 13 Ph.D. students, four of which have [25] , Lineinye˘ i kvazilineinye˘ uravneniya èllipticheskogo tipa, habilitated. Izdat. “Nauka”, Moscow, 1964. MR 0211073

10 0 [26] O. A. Ladyzhenskaia and N. N. Ural tzeva, On the smoothness [44] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, of weak solutions of quasilinear equations in several variables Acta Math. 138 (1977), no. 3-4, 219–240. MR 474389 0 and of variational problems, Comm. Pure Appl. Math. 14 (1961), [45] N. N. Ural ceva, Regularity of solutions of multidimensional el- 481–495. MR 149076 liptic equations and variational problems, Soviet Math. Dokl. 1 0 [27] O. A. Ladyzhenskaya and N. N. Ural tseva, Local estimates for (1960), 161–164. MR 0126742 gradients of solutions of non-uniformly elliptic and parabolic equa- [46] , Boundary-value problems for quasi-linear elliptic equations, Comm. Pure Appl. Math. 23 (1970), 677–703. MR 265745 tions and systems with principal part of divergence type, Dokl. [28] , Lineinye˘ i kvazilineinye˘ uravneniya èllipticheskogo tipa, Akad. Nauk SSSR 147 (1962), 313–316. MR 0142886 Izdat. “Nauka”, Moscow, 1973, Second edition, revised. MR [47] , General second-order quasi-linear equations and certain 0509265 classes of systems of equations of elliptic type, Dokl. Akad. Nauk [29] , A survey of results on the solvability of boundary value SSSR 146 (1962), 778–781. MR 0140817 problems for uniformly elliptic and parabolic second-order quasi- [48] , Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. linear equations having unbounded singularities, Uspekhi Mat. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184–222. Nauk 41 (1986), no. 5(251), 59–83, 262. MR 878325 MR 0244628 Nonlinear boundary value problems for equations of min- [30] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasi- [49] , imal surface type 116 linear elliptic equations, Academic Press, New York-London, , Trudy Mat. Inst. Steklov. (1971), 217–226, 1968, Translated from the Russian by Scripta Technica, Inc, 237, Boundary value problems of mathematical physics, 7. MR Translation editor: Leon Ehrenpreis. MR 0244627 0364860 The regularity of the solutions of variational inequalities [31] John L. Lewis, Regularity of the derivatives of solutions to certain [50] , , degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 no. 6, 849–858. MR 721568 (1972), 211–219, Boundary value problems of mathematical 0 physics and related questions in the theory of functions, 6. MR [32] A. I. Nazarov and N. N. Ural tseva, Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equa- 0313623 [51] , The solvability of the capillarity problem, Vestnik tion, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. Leningrad. Univ. (1973), no. 19 Mat. Meh. Astronom. Vyp. 4, (LOMI) 147 (1985), 95–109, 204–205, Boundary value problems 54–64, 152. MR 0638359 of mathematical physics and related problems in the theory of [52] , The solvability of the capillarity problem. II, Vestnik functions, No. 17. MR 821477 Leningrad. Univ. (1975), no. 1, Mat. Meh. Astronom. vyp. 1, The Harnack inequality and related properties of solu- [33] , 143–149, 191, Collection of articles dedicated to the memory of tions of elliptic and parabolic equations with divergence-free lower- Academician V. I. Smirnov. MR 0638360 order coecients 23 , Algebra i Analiz (2011), no. 1, 136–168. MR [53] , Strong solutions of the generalized Signorini problem, 2760150 0 Sibirsk. Mat. Zh. 19 (1978), no. 5, 1204–1212, 1216. MR 508511 Long time behavior of ows 0 [34] V. I. Oliker and N. N. Ural tseva, [54] N. N. Ural tseva, Estimates of the maximum moduli of gradi- moving by mean curvature , Nonlinear evolution equations, Amer. ents for solutions of capillarity problems, Zap. Nauchn. Sem. Math. Soc. Transl. Ser. 2, vol. 164, Amer. Math. Soc., Providence, Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 274– RI, 1995, pp. 163–170. MR 1334142 0 284, 312, Boundary value problems of mathematical physics and [35] Vladimir I. Oliker and Nina N. Ural tseva, Long time behavior of related questions in the theory of functions, 14. MR 660089 ows moving by mean curvature. II, Topol. Methods Nonlinear [55] , Hölder continuity of gradients of solutions of parabolic Anal. 9 (1997), no. 1, 17–28. MR 1483640 equations with boundary conditions of Signorini type, Dokl. Akad. [36] Vladimir I. Oliker and Nina N. Uraltseva, Evolution of nonpara- Nauk SSSR 280 (1985), no. 3, 563–565. MR 775926 metric surfaces with speed depending on curvature. II. The mean [56] , Estimation on the boundary of the domain of deriva- curvature case, Comm. Pure Appl. Math. 46 (1993), no. 1, 97–135. tives of solutions of variational inequalities, Linear and nonlinear MR 1193345 boundary value problems. Spectral theory (Russian), Probl. Mat. [37] , Evolution of nonparametric surfaces with speed depend- Anal., vol. 10, Leningrad. Univ., Leningrad, 1986, Translated in ing on curvature. III. Some remarks on mean curvature and J. Soviet Math. 45 (1989), no. 3, 1181–1191, pp. 92–105, 213. MR anisotropic ows, Degenerate diusions (Minneapolis, MN, 860572 1991), IMA Vol. Math. Appl., vol. 47, Springer, New York, 1993, [57] , Estimates of derivatives of solutions of elliptic and pp. 141–156. MR 1246345 parabolic inequalities, Proceedings of the International Congress [38] Arshak Petrosyan, Henrik Shahgholian, and Nina Uraltseva, of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Regularity of free boundaries in obstacle-type problems, Gradu- Soc., Providence, RI, 1987, pp. 1143–1149. MR 934318 ate Studies in Mathematics, vol. 136, American Mathematical [58] , On the regularity of solutions of variational inequalities, Society, Providence, RI, 2012. MR 2962060 Uspekhi Mat. Nauk 42 (1987), no. 6(258), 151–174, 248. MR [39] Rainer Schumann, Regularity for Signorini’s problem in linear 933999 1 elasticity, Manuscripta Math. 63 (1989), no. 3, 255–291. MR [59] , C regularity of the boundary of a noncoincident set in a 986184 problem with an obstacle, Algebra i Analiz 8 (1996), no. 2, 205– [40] Henrik Shahgholian and Nina Uraltseva, Regularity properties 221. MR 1392033 of a free boundary near contact points with the xed boundary, Duke Math. J. 116 (2003), no. 1, 1–34. MR 1950478 [41] Henrik Shahgholian, Nina Uraltseva, and Georg S. Weiss, The two-phase membrane problem—regularity of the free boundaries in higher dimensions, Int. Math. Res. Not. IMRN (2007), no. 8, Art. ID rnm026, 16. MR 2340105 [42] Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Dierential Equations 51 (1984), no. 1, 126– 150. MR 727034 [43] Neil S. Trudinger, Linear elliptic operators with measurable coecients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 265–308. MR 369884