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Home , Alessio Figalli, Isadore Singer, Jacques Tits, Lennart Carleson, Louis Nirenberg

Recent Applications of Nirenberg’s Classical Ideas Communicated by Christina Sormani

Figalli describe the stability of the GNS inequality and recent applications of this result. Mu-Tao Wang and Shing- Tung Yau describe applications of Nirenberg’s solution of the Weyl problem to general relativity. Of course there are many, many more important applications of Nirenberg’s work that we cannot touch on in this article. In fact Nirenberg has been cited over 10,000 times by over 5,000 authors. We hope just to give a flavor of the new directions being taken with his work. Xavier Cabré

The Gidas-Ni-Nirenberg Theorem In 1979, Gidas, Ni, and Nirenberg [4] established an impor- tant result on monotonicity and of solutions

Used with permission, Carol Hutchins CIMS. to nonlinear second order elliptic PDEs (see Figure 1):

Louis Nirenberg teaching at the Courant Institute of 푛 Mathematical Sciences, University. Theorem 1. Let Ω be a bounded smooth domain of ℝ = 푛−1 ′ ℝ×ℝ which is symmetric with respect to {(푥1, 푥 ) ∶ 푥1 = n 2015, and John F. Nash Jr. were 0}: awarded the “for striking and seminal if (푝 , 푥′) ∈ Ω, then the reflected point (−푝 , 푣) ∈ Ω contributions to the theory of nonlinear partial dif- 1 1 ferential equations and its applications to .” While many an article has been written describingI Nirenberg’s greatest works, here we have asked to describe how they have applied Nirenberg’s ideas in new and exciting ways. We begin with Xavier Cabré, a former student of Niren- berg, who describes extensions of the Gidas-Ni-Nirenberg results on the of solutions to nonlinear elliptic partial differential equations. Alice Chang then describes recent work on Nirenberg’s problem: prescribing the Gauss curvature on a sphere. Gregory Seregin discusses recent work on the Navier-Stokes problem applying the work of Caffarelli-Kohn-Nirenberg. Eric Carlen and Alessio

Christina Sormani is professor of at Lebman Col- lege and CUNY Graduate Center. Her email address is sormanic@ gmail.com. Xavier Cabré is professor of mathematics at the University of Politecnica de Catalunya. His email address is xavier.cabre@upc. edu. Figure 1. Students at the City University of New York For permission to reprint this article, please contact: discussing the symmetries of solutions as described [email protected]. in the Gidas-Ni-Nirenberg Theorem. DOI: http://dx.doi.org/10.1090/noti1332

2 Notices of the AMS Volume 63, Number 2 and is convex in the 푥1-direction: ′ ′ if 푝 = (푝1, 푥 ) ∈ Ω and 푞 = (푞1, 푥 ) ∈ Ω, then the line segment 푝푞 ⊂ Ω. Let 푓 be a locally Lipschitz function and 푢 be a bounded positive solution of (1) − Δ푢 = 푓(푢) in Ω with 푢 = 0 on 휕Ω.

Then 푢 is symmetric with respect to {푥1 = 0}: ′ ′ ′ 푢(푝1, 푥 ) = 푢(−푝1, 푥 ) ∀(푝1, 푥 ) ∈ Ω, and 푢 is monotone for 푥1 > 0: Public Domain. 휕 푢 < 0 in Ω ∩ {푥 > 0}. 푥1 1 Sophus Lee and Niels Henrik Abel, the two Norwegian In particular, if Ω = 퐵푅 is a ball, then 푢 is radially symmet- mathematicians who inspired the Abel Prize. ric and

휕푟푢 < 0 ∀푥 ∈ 퐵푅\{0} where 푟 = |푥|. The proof uses the maximum principle and is very flexible. It allows for extensions to some unbounded The proof of the Gidas-Ni-Nirenberg result uses the domains (in particular, all of ℝ푛), to fully nonlinear elliptic powerful Alexandrov moving planes method. Alexandrov equations, to some nonlinearities 푓(푥, 푢) depending also developed this method circa 1955 in a series of papers on the variable 푥 ∈ Ω, as well as to some elliptic systems to establish (among other results) that spheres are the of equations. As a consequence, this is Louis Nirenberg’s only embedded, bounded, and connected hypersurfaces most cited paper. As he once said, “I made a living off the 푛 maximum principle.” of ℝ with constant mean curvature. In 1971, J. Serrin If for a certain nonlinearity function, 푓, one knows [6] used the method to prove that balls are the only that there is uniqueness of positive solution to the bounded smooth domains that admit a solution to certain above boundary value problem (this happens for instance overdetermined boundary value problems. when 푓 is nonincreasing, since then the energy func- In the Alexandrov method, one considers the planes tional is convex), then the solution 푢 must be symmetric 푇휆 = {푥1 = 휆}, the domains Σ휆 = Ω ∩ {푥1 > 휆}, and the as an immediate consequence of the uniqueness. How- 휆 ′ reflected function 푢 (푥) = 푢(2휆 − 푥1, 푥 ) (the reflection ever, in many interesting applications (ground states in 휆 of 푢 across 푇휆) as in Figure 2. One shows that 푢 < 푢 in mathematical physics, curvature equations in conformal ∗ geometry), uniqueness does not hold, and the symme- Σ휆 for all 0 < 휆 < 휆 ∶= supΩ 푥1. One starts proving this ∗ try result requires a proof. Note also that positivity of for 휆 close to 휆 and then showing, through a continuity the solution is needed in the assumptions. Indeed, the argument, that the same holds all the way down to 휆 = 0. second Dirichlet eigenfunction of the Laplacian in a ball The two ingredients used here are the Hopf boundary (like sin(푥) on [−휋, 휋]) satisfies −Δ푢 = 휆2푢, but it is not lemma and the strong maximum principle. They are radially symmetric. In fact, it is odd with respect to one applied to the difference 푢휆 − 푢, which satisfies a linear hyperplane through the center of the ball. equation, since 푢휆 solves the same nonlinear equation as the one for 푢.

Sidebar 1. Abel Prize Winners x 2 2015: John Forbes Nash, Jr. and Louis Nirenberg Ω 2014: Σλ 2013: x1 2012: Endre Szemerédi 2011: 2010: 2009: Mikhail Leonidovich Gromov

Tλ 2008: John G. Thompson and 2007: S. R. Srinivasa Varadhan 2006: Figure 2. In the Alexandrov method, one reflects 2005: domains Σ휆 across vertical hyperplanes 푇휆 that move continuously from the right to the middle. 2004: and 2003: Jean-Pierre Serre

February 2016 Notices of the AMS 3 Sidebar 2. Notices articles on Nirenberg.

Nash and Nirenberg Awarded 2015 Abel Prize, June/ July, 2015 On the Work of Louis Nirenberg by , March, 2011 Interview with Louis Nirenberg, April, 2002 Louis Nirenberg Receives National Medal of Science, October, 1996

The use of the Hopf boundary lemma required some regularity of the domain Ω in the original paper of Gidas- Xavier Cabré with Louis Nirenberg in 2015. Ni-Nirenberg, and thus symmetry in domains such as a square in the plane was left open. A new approach was then developed by Berestycki and Nirenberg in 1991. measure, to obtain the Gidas-Ni-Nirenberg type symmetry It replaced the use of the Hopf lemma by a maximum results for locally Lipschitz nonlinearities 푓. In particular, principle in “small domains”, domains such us Σ휆 when 휆 if Ω is a ball, we proved that any solution 푢 to (2) is ∗ is very close to 휆 or Σ휆\퐾, where 휆 is arbitrary and 퐾 is a rotationally symmetric and 푢푟 < 0 within the ball. sufficiently large compact set inside Σ휆. In this way, they succeeded in proving symmetry in nonsmooth domains References such as cubes. [1] H. Berestycki and L. Nirenberg, On the method of moving In addition, Berestycki and Nirenberg introduced the planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 sliding method: a new and powerful tool leading to the (1991), no. 1, 1–37. monotonicity of solutions in certain problems (and some- [2] M. Birkner, J. A. López-Mimbela, and A. Wakolbinger, times to their uniqueness or to their oddness). It consists Comparison results and steady states for the Fujita equa- of comparing the solution 푢 with the slided (or translated) tion with fractional Laplacian, Ann. Inst. H. Poincaré Anal. ′ Non Linéaire 22 (2005), 83–97. solutions 푢휆(푥) ∶= 푢(푥1 + 휆, 푥 ). Within elliptic PDEs there has been great activity in [3] X. Cabré and Jinggang Tan, Positive solutions of nonlin- these last years on equations involving fractional Lapla- ear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052–2093. cians or related nonlocal operators, for instance, the [4] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and re- generators of Lévy stable diffusion processes. They arise lated properties via the maximum principle, Comm. Math. when studying anomalous diffusions in plasmas, chemical Phys. 68 (1979), no. 3, 209–243. reactions in liquids, geophysical fluid dynamics, popula- [5] J. Serrin, A symmetry problem in , Arch. tion dynamics, and finance. The Gidas-Ni-Nirenberg result Rational Mech. Anal. 43 (1971), 304–318. has recently been extended to many of these fractional operators. For example, Birkner, López-Mimbela, and Sun-Yung Alice Chang Wakolbinger extended it to the Dirichlet problem in a ball 푠 for the fractional Laplacian, which reads (−Δ) 푢 = 푓(푢) Nirenberg’s Problem in 퐵 and 푢 = 0 on ℝ푛\퐵 (the complement of 퐵 ). 푅 푅 푅 In 1970, Louis Nirenberg proposed the problem of “pre- In 2010, Jinggang Tan and I [3] used the moving scribing Gaussian curvature on the sphere.” This question planes method to show symmetry and monotonicity for has led to one of the most active and exciting subfields the spectral square root of the Laplacian on a bounded of geometric analysis. The penetrating question raised by domain with zero Dirichlet boundary conditions. That is, Nirenberg is, What are the functions, 퐾, which can arise as we considered positive solutions to the Gaussian curvature function, 퐾푔, for some Riemannian (2) 퐴1/2푢 = 푓(푢) in Ω with 푢 = 0 on 휕Ω. metric, 푔, on the sphere? On the standard two-dimensional sphere, by the Gauss- To define 퐴1/2, we took the eigenvalues, 휆푘, and eigen- Bonnet formula, one has functions, 휑푘, of the Laplacian in Ω:

−Δ휑푘 = 휆푘휑푘 in Ω with 휑푘 = 0 on 휕Ω. (3) ∫ 퐾푔푑푣푔 = 4휋. 푆2 Then, given the expansion 푢 = ∑∞ 푐 휑 , we set 퐴 푢 = 푘=1 푘 푘 1/2 Thus any such Gaussian curvature function 퐾 must be ∑∞ 푐 휆1/2휑 . 푔 푘=1 푘 푘 푘 positive somewhere on the sphere. See Figure 3 for a This is a nonlocal problem, but we transformed it into surface with positive Gaussian curvature everywhere and a local problem in one more dimension by viewing it as this issue’s article “What Is Gauss Curvature?” (page X) a Dirichlet to Neumann operator: given 푢 in Ω find its for more information about Gaussian curvature. harmonic extension 푤 in Ω × [0, ∞) with zero Dirichlet 휕푤 condition on 휕Ω × [0, ∞). Then we have 퐴1/2푢 = 휕휈 |Ω×{0}. Sun-Yung Alice Chang is Eugene Higgins Professor of Mathe- We were then able to apply the moving planes method, matics at . Her email address is chang@ combined with a maximum principle in domains of small math.princeton.edu.

4 Notices of the AMS Volume 63, Number 2 At the time that Nirenberg’s problem was originally posed, it was considered that this might be the only restriction on the function 퐾 = 퐾푔. Then in 1974, Kazdan- Warner ([4]) indicated that this is not the case, that 퐾 = 퐾푔 must satisfy another condition:

(4) ∫ ⟨∇퐾, ∇푥푖⟩푑휇푔 = 0, 푆2 where 푥푖, 푖 = 1, 2, 3, are the coordinate functions of the sphere. In particular, the Nirenberg problem has no solution for 퐾(푝) = 푓(푥1(푝)) where 푓 is monotone. Nirenberg’s problem was first studied in the 1971 doctoral of D. Koutroufiotis, a PhD student of Nirenberg. Here 푔 was assumed to be conformal to the 2 standard metric, 푔0, on the two-dimensional sphere, 핊 . 2휔 If one writes 푔 = 푒 푔0, then the problem is equivalent to solving (5) − Δ휔 + 1 = 퐾푒2휔 on 핊2, Louis Nirenberg, Alice Chang and Paul Yang in 2015. where Δ is the Laplacian with respect to the standard 2 metric 푔0 on 핊 . Koutroufiotis proved that when 퐾 is antipodally symmetric and sufficiently close to 1, then there is a solution, 푔, to the problem 퐾 = 퐾푔. In 1973 Moser proved that for any antipodally sym- is the Gauss curvature 퐾푔 of a metric 푔 conformal to 푔0. metric function 퐾 which is positive somewhere, there is a In joint work with Gursky and Yang appearing in 1993, solution to this original version of the Nirenberg problem. the author provided an “index formula” for 퐾 to be the This important breakthrough was founded upon Moser’s Gaussian curvature of a metric 푔 conformal to 푔0. In 2005, inequality, a sharp version of a limiting Sobolov imbed- Michael Struwe re-proved our 1987 theorem using flow ding result by Trudinger. To prove his result, Moser first methods and produced an example demonstrating that observed that solutions 휔 to (5) are the critical points of the hypotheses are in some sense the best possible. More the functional precisely, Struwe constructed functions, 퐾, having exactly 1 2휔 1 2 two local maxima and one saddle point, 푝, where Δ퐾(푝) < 퐹퐾[휔] = log ∫ 퐾푒 푑휇0− ∫ (|∇휔| + 2휔) 푑휇0. 4휋 핊2 4휋 핊2 0, which cannot be realized as curvature functions of conformal metrics on the two-dimensional sphere. Moser’s inequality provides a bound on the terms in the In another direction, the problem of prescribing right-hand side of this inequality. One may then find a Gaussian curvature further developed into the prob- solution to the Nirenberg problem as a limit of a sequence of functions via a minimax procedure using this bound lem of “prescribing 푄-curvature”. The connection of to prove the sequence converges. This method has led 푄-curvature to the Gauss curvature was first pointed out to many additional partial solutions of the Nirenberg by T. Branson in 1995. For example, on 4-spheres, through 2 problem, including results by the author with P. Yang, the Gauss-Bonnet formula, one has ∫푆4 푄푔푑푣푔 = 32휋 . It K. C. Chang with J. Q. Liu, Z. C. Han, and others. turns out that 푄-curvatures on compact manifolds of even In particular, the author and Paul Yang proved the dimensions share many of the same analytic properties following theorem in 1987: Let 퐾 be a positive smooth as the Gaussian curvature on compact surfaces, and the function with only nondegenerate critical points. Suppose tool introduced by Moser can be effectively applied. Study there are at least two local maxima of 퐾 and suppose that of 푄-curvature has become an important topic in confor- at all saddle points, 푝, of 퐾, we have Δ푔0 퐾(푝) > 0. Then 퐾 mal geometry, with applications to problems in Ads/CFT theory (by Graham-Zworski and Fefferman-Graham) and geometry (by Chang-Gursky-Yang and others). The field is active and developing (with important papers in 2013 by Branson-Fontana-Morpurgo and by Case-Yang).

On a Personal Note: My relationship with Louis began as that of a mentor and later a respected, well-admired friend. All the time I have spent with him—either discussing mathematics or just chatting—has been wonderful. It is like the saying in Figure 3. An ellipsoid shaded by Gaussian curvature. Chinese, “You are enjoying the breeze of the spring,” a wonderful experience in life.

February 2016 Notices of the AMS 5 6 Col- Hilda’s is address St. email at His ox.ac.uk Oxford. mathematics of of University professor lege, is Seregin Gregory space. in measure Hausdorff Scheffer one-dimensional finite 1970s of solutions, vorticity, the such the would that of In uniqueness such existence their smoothness. the 1934, as their proved in and, from Leray globally, follow exist J. and by Such energy observed solutions. finite Leray-Hopf have weak solutions so-called the consider 4. Figure See pressure. the is of ∞) [0, × value 휕Ω the with initial equations: Navier-Stokes the mathematical (6) the of of for problem well-posedness field value the global boundary in a problems is main hydrodynamics the of One Problem Navier-Stokes the and Caffarelli-Kohn-Nirenberg Seregin Gregory [6] [5] [4] [3] [2] [1] References 2015. in at ceremony (right) Prize Caffarelli AMS Luis the and (center) Nirenberg Louis 휕

푡 Photo by Sandy Huffakel. 푢−Δ 푝=푓 = ∇푝 + Δ푢 − ∇푢 ⋅ 푢 + 푢 n ftepsil ast takti usini to is question this attack to ways possible the of One ah J. Math. Struwe M. 273–280. pp. 1973, NY, Press, Symposium Systems Dynamical scalar and Moser J. Gaussian prescribed with curvature, metrics of formation Kazdan J. Fefferman C. Chang on A. curvature Y. S. 3671–3742. (1995), series, complementary the and Branson T. metrics, . 128 ah e.Lett Res. Math. nannierpolmi ieeta geometry, differential in problem nonlinear a On , Here . n.o Math. of Ann. o praht iebr’ problem, Nirenberg’s to approach flow A , and hr nqaiy h ucinldeterminant functional the inequality, Sharp , 20) o ,19–64. 1, no. (2005), 푆 2 and , 푢 caMath. Acta .Warner F. and stevlct edo h udand fluid the of field velocity the is .R Graham R. C. 푢 푢 × ∇ rsrbdon prescribed . .C Yang C. P. 9 101 20) o.2ad3 139–152. 3, and 2 nos. (2002), 159 n div and xsec n ofra de- conformal and Existence , scniuu usd set a outside continuous is , 17)317–331. (1975) rn.Ae.Mt.Soc. Math. Amer. Trans. 18) 215–259. (1987), M ext,e. Academic ed.) Peixoto, (M. , 푄 craueadPoincaré and -curvature rsrbn Gaussian Prescribing , 0 = 푢 {0} × Ω seregin@maths. 푢 on o l time, all for , 0 ∞), [0, × Ω oie fteAMS the of Notices n on and Duke 347 푝 cylinders: 휆 hnoemydfieteoedmninlHausdorff one-dimensional the cylinders, parabolic define these using may measure one Then eksltosaesot wyfo iglrset on singular solution weak a suitable a from be to away smooth 푆 are solutions weak and fteflwcnb on ikdfo i webpage his from linked found be devuyst.perso.math.cnrs.fr can videos flow Complete the Vuyst. of de Florian of by implementation Navier-Stokes numerical A 4. 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(푥 휆 푝(휆푥, (,푡) 푓(푥, 2 2 푦+ 푑푦 푡) lostsy()for (6) satisfy also then , 푄 . 푟 oue6,Nme 2 Number 63, Volume ∫ (푥,푡) 푄 egadhsco-authors his and space-time berg Niren- Louis in domains. tions equa- Navier-Stokes to the suit- solutions of weak able notion new of a introduction ex- their plicit is paper of this novelty conceptual The fluid. compressible in- viscous a of flow the of deter- description ministic a equa- provide the tions not Navier-Stokes or understanding whether to- most the ward steps of important one is paper Kohn-Nirenberg 푢 푟 푥 푡) (푥, h 92Caffarelli- 1982 The 휆 |∇푢| 푥 )=휆(푥 휆 휆푢(휆푥, = 푡) (푥, (,푡) 푢(푥, 푃 2 2 if 푦휏<∞, < 푑푦푑휏 1 푡}. < 휏 < . and 푓 휆 푥 )= 푡) (푥, (,푡) 푝(푥, (,푡) 푢(푥, 2 푡) if 푝 ∈ 퐿 5 (푄푟(푥, 푡)), and if 푢 and 푝 satisfy the classical earlier paper, one considers weak Leray-Hopf solutions 4 Navier-Stokes system in the sense of distributions on to Navier-Stokes equations on ℝ3 × [0, 푇) with initial data 푄푟(푥, 푡): 푢(푥, 0) = 푣0(푥) smooth with compact support. If the finite energy solution arising from 푣0 blows up at time 푇 < ∞, 2 2 ∫ |∇푢| 휑푑푦푑휏 then 푄 (푥,푡) 푟 lim ||푢(푥.푡)||퐿3(ℝ3) = ∞. 푡→푇− 2 2 ≤ ∫ [|푢| (휕휏휑 + Δ휑) + 푢 ⋅ ∇휑(|푢| + 2푝)]푑푦푑휏 In order to prove this result, one needs to apply a 푄푟(푥,푡) backward uniqueness theorem for a certain parabolic for any nonnegative smooth function 휑 compactly sup- differential inequality. We need to provide a decay at the ported in 푄푟(푥, 푡). The space for pressure can be taken spatial infinity for some ancient (backward) solution to the slightly differently in order to simplify statements, and Navier-Stokes equations. Recall that an ancient solution such a space is 퐿 3 (푄푟(푥, 푡)). With this minor simplifica- is one which is well defined for all time 푡 ∈ (−∞, 0) 2 tion and the assumption 푓 = 0, one important theorem in and is found by taking a parabolic rescaling around the Caffarelli-Kohn-Nirenberg may be restated as follows: developing singularity. In particular, having in our hands a finite global 퐿3-norm, we can make the tails of the Theorem 2. If 푢 and 푝 are a suitable weak solution to the corresponding integrals small enough and then apply the Navier-Stokes equations in 푄푟(푥, 푡), then there exist posi- Caffarelli-Kohn-Nirenberg theorem stated above. tive universal constants 휀 and 푐푘, 푘 = 0, 1, … , such that In a sense, the results of the Caffarelli-Kohn-Nirenberg if paper seem to be optimal for a local solution. To go 1 3 3 ∫ (|푢| + |푝| 2 )푑푦푑휏 < 휀, further, one needs additional information on the global 푟2 푄푟(푥,푡) level, i.e., on the level of initial boundary value problems for the Navier-Stokes system. In his interview with the then |∇푘푢(푦, 휏)| ≤ 푐 /푟푘+1 for any (푦, 휏) ∈ 푄̄ (푥, 푡) and 푘 푟/2 Notices of AMS in 2002, Nirenberg pointed out that further any 푘 ∈ ℕ. applications of could be useful toward In fact, the linear theory implies that all functions ∇푘푢 making additional advances. are Hölder continuous in the closure of 푄푟/2(푥, 푡). The exponent of Hölder continuity is related to the choice of References a functional space for the pressure. [1] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regular- Let me give a simple application of Theorem 2 that arose ity of suitable weak solutions of the Navier-Stokes equations, when proving the Ladyzheskaya-Prodi-Serrin condition of Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. regularity by rescaling around a possible singular point. [2] L. Escauriaza, G. Seregin, and V. Šverák, 퐿3,∞-solutions of After a certain rescaling, there are sufficiently smooth Navier-Stokes equations and backward uniqueness (Russian), functions 푢 and 푝 that satisfy the Navier-Stokes equations Uspekhi Mat. Nauk. 58 (2003), no. 2(350), 3–44; translation in 3 Russian Math. Surveys 58 (2003), no. 2, 211–250. in ℝ ×] − ∞, 0[. Suppose that ‖푢‖ 3 is bounded. 퐿5(ℝ ×]−∞,0[) [3] J. Leray, Sur le mouvement d’un liquide visqueux emplissant Let us prove a Liouville-type theorem saying that 푢 = 0. l’espace, Acta Math. 63 (1934), 193–248. Given 휇 > 0, we can find 퐴 < 0 such that [4] G. Seregin, A certain necessary condition of potential blow 퐴 up for Navier-Stokes equations, Comm. Math. Phys. 312 5 5 ∫ (|푢| + |푝| 2 )푑푥푑푡 < 휇. (2012), no. 3, 833–845. −∞

Fix any 푧0 = (푥0, 푡0) with 푡0 ≤ 퐴. Then for any 푅 > 0, Eric Carlen and Alessio Figalli

1 3 3 ∫ (|푢| + |푝| 2 )푑푥푑푡 푅2 Stability of the Gagliardo-Nirenberg Sobolev 푄푅(푧0) Inequality 3 5 5 5 2 ≤ 푐( ∫ (|푢| + |푝| 2 )푑푥푑푡) ≤ 푐휇 5 . A central focus of Nirenberg’s research has been partial differential equations. An important tool in their study 푄 (푧 ) 푅 0 consists of using functional inequalities that control 2 푞 If 휇 5 ≤ 휀, then one can use Theorem 2 and conclude some 퐿 norm of a function in terms of the norm of its that |푢(푧0)| ≤ 푐0/푅. Taking 푅 → ∞, we can deduce that derivatives. 푢 = 0 in ℝ3×] − ∞, 퐴[. We then split the interval ]퐴, 0[ The Sobolev inequalities are the classic example. Let into sufficiently small pieces and sequentially exclude the 1/푞 푞 corresponding layers in order to complete the proof that ‖푢‖푞 = (∫ |푢(푥)| d푥) , 푢 = 0. ℝ푛 A more sophisticated application of this theorem appears in my 2012 paper on the potential blowup for Eric Carlen is professor of mathematics at Rutgers University. His Navier-Stokes equations and in an earlier paper of mine email address is [email protected]. with Escauriaza and Šverák appearing in 2003. In the Alessio Figalli is professor of mathematics at the University of Texas more recent paper, which improves upon the work in the at Austin. His email address is [email protected].

February 2016 Notices of the AMS 7 4 motivated work to determine the sharp constants and 3.5 optimizers. While it is relatively easy to show that minimizers exist 3 and are radially symmetric, finding the sharp constants

2.5 and the explicit form of the minimizers has only been done in a few cases. Carlen and Loss [2] did this for the Nash 2 3 0 inequalities, and Del Pino and Dolbeault [3] did this for 2.5 0.5 2 1 the one-parameter family of inequalities corresponding to 1.5 1.5 the case 푝 = 2, 푞 = 푡 + 1, and 푟 = 2푡. The work in [2] was 1 2 0.5 2.5 motivated by a question raised by Kato concerning the 0 3 2-dimensional Navier-Stokes equation, while the authors Figure 5. The Keller-Segel model of chemotaxis shows of [3] used the sharp form of these GN inequalities to the characteristic clumping. Numerical study nonlinear diffusions. implementation by Ibrahim Fatkullin. Recent works have shown that going beyond the characterization of the optimizers and proving a stability result can actually give further information on the rate of 푞 for 1 ≤ 푞 < ∞, denote the 퐿 norm of a function 푢. Then, approach to the equilibrium for such nonlinear diffusion for 1 ≤ 푝 < 푛, there exists a constant 퐶(푛, 푝) such that equations. For example, consider the Gagliardo-Nirenberg 푛 for all smooth functions 푢 ∶ ℝ → ℝ vanishing at infinity, in the particular case 푛 = 2, 푝 = 2, ∗ 푛푝 푟 = 4, 푠 = 6. In this case, the sharp form found by Del Pino ‖푢‖ ∗ ≤ 퐶(푛, 푝)‖∇푢‖ , 푝 ∶= . 푝 푝 푛 − 푝 and Dolbeault takes the form ∗ The value of 푝 is determined by scaling: inserting 푢(휆푥), 휋 ‖푢‖6 ≤ ‖∇푢‖2‖푢‖4, ∗ 6 2 4 휆 > 0, into the inequality above, the 퐿푝 norm is the only and there is equality if and only if, up to a dilation, one for which both sides are proportional to the same 푢(푥) = 푣 (푥) ∶= 휆1/3(1 + 휆2|푥 − 푥 |2)−1/2. power of 휆. 휆,푥0 0 We now define the quantity As useful as these inequalities are, in the late 1950s 2 4 6 mathematicians studying elliptic and parabolic regularity 훿퐺푁[푢] ∶= ‖∇푢‖2‖푢‖4 − 휋‖푢‖6. understood the need for an extension of the Sobolev Note that by the above, 훿 [푢] is nonnegative. Further- inequalities. The first example in this direction appeared 퐺푁 more, if 훿 [푢] = 0, then 푢 is an optimizer. We say this in Nash’s famous paper about the regularity of solu- 퐺푁 last fact is “stable” if 훿 [푢] < 휀 for 휀 > 0 sufficiently tions to parabolic equations with measurable coefficients 퐺푁 small implies that 푢 is close to an optimizer in some [5], where he showed what is now called the “Nash suitable norm. inequalities”:1 In 2013 we proved the following quantitative stability 2 푛/(푛+2) 2/(푛+2) ‖푢‖2 ≤ 퐶푛 ‖∇푢‖2 ‖푢‖1 . estimate: Basically at the same time as Nash, both Gagliardo Theorem 4. Let 푢 ∶ ℝ푛 → ℝ be a smooth nonnegative and Nirenberg independently proved a general family of function vanishing at infinity such that ‖푢‖6 = ‖푣1,0‖6. inequalities including both the Sobolev inequalities and Then there exist universal constants 퐾1, 훿1 > 0 such that the Nash inequality as special cases. In their form, these whenever 훿 [푢] ≤ 훿 , can be stated as follows. 퐺푁푆 1 6 6 푛 (7) inf ‖푢 − 푣휆,푥 ‖1 ≤ 퐾1 훿퐺푁[푢]. Theorem 3 (Gagliardo-Nirenberg). Let 푢 ∶ ℝ → ℝ be a 2 0 √ 휆>0,푥0∈ℝ smooth function vanishing at infinity. Then, for any 1 ≤ 푝 ≤ 푛 and 1 ≤ 푟 ≤ 푞 ≤ 푝∗, In the same paper, we applied this result to obtain 휃 1−휃 a quantitative bound to the rate of convergence of ‖푢‖푟 ≤ 퐶(푛, 푝, 푟, 푠)‖∇푢‖푝‖푢‖푞 , equilibrium for the critical mass Keller-Segel equation. where 휃 ∈ [0, 1] satisfies 1/푟 = 휃 (1/푝 − 1/푛)+(1−휃)/푞 . This equation was introduced by Evelyn Keller and Lee Segel in 1970 to model the aggregation exhibited Again, the relation beween the exponents and 휃 is governed by scaling. Note that the Gagliardo-Nirenberg by slime molds, a name for several types of eukaryotic Sobolev inequalities hold and were proved by them for cells that, depending on environmental circumstances higher derivatives of 푢 and a larger set of exponents as and life cycle, either live freely as individual cells or else well. Here we stated only the most frequently used cases. aggregate into tightly bound units. The mechanism driving The initial applications of these inequalities to regu- aggregation in a population of free cells is chemotaxis: the larity problems did not require knowledge of the sharp cells emit a chemical attractant that draws the population constant 퐶(푛, 푝, 푟, 푠). However, the Gagliardo-Nirenberg together, while the diffusive motion of free cells promotes Sobolev inequalities have since found many other ap- dispersion of the population (see Figure 5). The equation plications for which this knowledge is useful and has is mathematically and biologically interesting because there is a critical balance between the dissipative effects 1In his paper, Nash thanks Elias Stein for the proof (using the of diffusion and the accretive effects of chemotaxis, and Fourier transform) of this inequality. a sharp Gagliardo-Nirenberg-Sobolev inequality together

8 Notices of the AMS Volume 63, Number 2 with related inequalities determines what happens at the threshold where these effects are balanced. It is interesting that the papers of Gagliardo, Nirenberg, and Nash all appeared in 1958, and their utility for questions concerning rates of smoothing and decay of norms for solutions of parabolic equations, which has been the source of much of their continuing interest, was already realized at this time. A posteriori, these results had a tremendous impact in surprisingly many applications and have inspired—and still do inspire—an enormous number of authors. Mu-Tao Wang presenting at MSRI in 2014. The References complete video of this introductory lecture on [1] Eric A. Carlen and Alessio Figalli, Stability for a GNS in- quasi-local mass is available at equality and the log-HLS inequality, with application to the https://www.msri.org/workshops/691. critical mass Keller-Segel equation, Duke Math. J. 162 (2013), no. 3, 579–625. [2] Eric A. Carlen and Michael Loss, Sharp constant in Nash’s inequality, Int. Math. Res. Notices, Vol. 1993, 1993, 213–215. been a central topic since the works of Hawking and [3] Manuel Del Pino and Jean Dolbeault, Best constants for Penrose. The idea of defining mass in general relativity Gagliardo-Nirenberg inequalities and applications to nonlin- went all the way back to Einstein. It was difficult because ear diffusions, J. Math. Pures Appl. (9) 81 (2002), no. 9, the theory of general relativity is a nonlinear theory where 847–875. gravity is dictated by a tensor, while Newtonian gravity is [4] Emilio Gagliardo, Proprietà di alcune classi di funzioni in basically linear and depends on a scalar. più variabili (Italian), Ricerche Mat. 7 (1958), 102–137. The concept of the mass of an isolated gravitational [5] John Nash, Continuity of solutions of parabolic and elliptic system was defined by Arnowitt-Deser-Misner (ADM) in equations, Amer. J. Math. 80 (1958), 931–954. [6] Louis Nirenberg, On elliptic partial differential equations, 1962 as an asymptotic value of the gravitational flux Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. through spheres expanding out to infinity. As a special case, one may assume the spacelike manifold is time symmetric and asymptotically flat and satisfies the dom- Mu-Tao Wang and Shing-Tung inant energy condition. So it is an asymptotically flat Yau three-dimensional with nonnega- tive scalar curvature. The positive mass theorem of the Isometric Embeddings of Surfaces second author and Schoen, proven in 1979, states that In 1916, Weyl proposed the conjecture that every closed the ADM mass of such a manifold is nonnegative and if surface with positive Gauss curvature can be isometri- the ADM mass is zero, then the manifold is flat Euclidean 2 cally embedded into Euclidean 3-space. Moreover, this 3-space. embedding should be unique up to an isometry of the A notion of quas-ilocal mass still needs to be defined Euclidean 3-space. Weyl’s problem can be considered as for bounded regions in a manifold. In fact, in the special an existence and uniqueness problem for a global elliptic year in geometry of 1979 at the Institute for Advanced equation, and he contributed an important estimate of Study, Penrose announced that the first major problem the second fundamental form. This was a major global in classical general relativity was the concept of quasi- problem in classical geometry in the first half of the local mass. Ideally such a notion should take on positive twentieth century. Lewy solved Weyl’s problem for real values on a three-dimensional manifold with nonnegative analytic metrics in 1932. scalar curvature unless the interior of the region is flat. It was not until 1954 that Nirenberg and Pogorelov It should asymptotically approach the ADM mass for solved Weyl’s problem for smooth metrics independently spheres expanding to infinity. It should depend only and using very different methods. While some of their upon the local properties of the boundary surface of the work depends on the works of Morrey in 1938 on the region, much as a flux depends only upon such properties. general theory of elliptic equations, these are spectacular One notion of quasi-local mass proposed by Brown- papers. Nirenberg’s paper also includes a solution to the York in 1991–93 applies the work of Nirenberg-Pogorelov. Minkowski problem. Both results have inspired much of The Brown-York quasi-local mass of a region is found the later work on global elliptic problems. by taking the difference of the total mean curvature of The work of Nirenberg-Pogorelov also has important the surface’s unique isometric embedding into Euclidean applications in general relativity. The question of defining three-dimensional space and subtracting the total mean and understanding the concept of quasi-local mass had curvature of the surface as it lies in the spacelike manifold. Brown-York made this important step towards the defini- Mu-Tao Wang is assistant professor of mathematics at Columbia tion of quasi-local mass by finding a surface Hamiltonian University. His email address is [email protected]. expression, which was derived from the Hilbert-Einstein Shing-Tung Yau is professor of mathematics at Harvard Uni- versity. His email address is [email protected]. 2For simplicity we discuss only three dimensions here.

February 2016 Notices of the AMS 9 functional. Their original version of the definition ap- With this positivity, we minimize the energy with peared to be gauge dependant (as it depended upon the respect to different time functions and define the quasi- three-manifold the surface enclosed). Many important local mass as the energy at critical points. This is very works appeared immediately, including a key paper of much like the rest mass of special relativity which realizes Hawking-Horowitz in 1996. the minimal energy seen among all observers. The Euler- Shi and Tam proved in 2002 that the Brown-York Lagrange equation is a fourth-order nonlinear equation of mass of a region in a three-dimensional manifold with the time function. Together with the isometric embedding nonnegative scalar curvature is nonnegative. Moreover, equation, this gives a well-determined system of four if the Brown-York mass is zero, then the region is iso- equations for four unknowns. This work was published metric to the region lying within the Nirenberg-Pogorelov in 2009. embedding into Euclidean space. It is interesting to note We do not yet have a general existence or uniqueness that Shi-Tam were unaware of the work of Brown-York theorem for this system. However, when the quasi-local and the applications to general relativity at the time and mass is positive, the system is elliptic and the linearized were concerned only with manifolds of nonnegative scalar equation can be solved. This is sufficient for studying curvature and the work of Nirenberg-Pogorelov. many unsettled problems for isolated gravitating systems, In general relativity it is essential to study a wide on which most current study of general relativity focuses. class of spacelike manifolds, not just those which are We expect more applications to come when this optimal time symmetric and satisfy the positive energy condition isometric embedding is better understood. (and thus have positive scalar curvature). In general such spacelike manifolds in spacetime have both a metric On a Personal Note by M. T. Wang tensor and a second fundamental form. They may be I was once invited to Professor Nirenberg’s apartment viewed as initial data sets, and indeed Choquet-Bruhat more than ten years ago. I explained my work on higher proved in 1952 the existence and uniqueness of solutions codimensional mean curvature flows and minimal surface to Einstein’s equation given such initial data. A notion of systems to him. He was quite surprised at first and was quasi-local mass of a region in such a manifold should very attentive to the detail of my explanation. Overall, it depend only upon this data on the boundary surface of was a very encouraging and inspiring experience for me. the region. It should be zero when the manifold is a spacelike submanifold of flat Minkowski space (which has On a personal note by S. T.-Yau no matter field and no gravitation). In 1974, Calabi and Niren- Liu-Yau introduced a quasi-local mass that was proven berg announced that they to be positive using Shi-Tam’s proof and both Schoen- had solved the bound- Yau’s proof and Witten’s proof of the positive mass ary value problem for the theorem. However, a simple calculation shows that a real Monge-Ampère equa- surface in the light cone of the Minkowski spacetime has tion. S. Y. Cheng and I also strictly positive Brown-York and Liu-Yau mass unless it thought we had a solution. is a round sphere. Chern told us that he was Examining this problem together, it became clear to us told that there was an error that isometric embeddings into the Minkowski spacetime in the Calabi-Nirenberg solu- need to be taken into account in order to obtain a mass tion, so he met with us and expression that satisfies the last criterion. By a simple with Nirenberg to go over Photo by Susan Towne Gilbert. degree of freedom counting, this yields an underdeter- our proof. Still it turned out Shing-Tung Yau. mined problem, and an additional equation is needed to the boundary estimates were not strong enough to solve expect any type of uniqueness. On the other hand, the sur- the problem. Three years later I told Nirenberg that face Hamiltonian expression relies on a definite choice Cheng and I could solve the problem without the strong of gauge choice, without which the definition remains boundary estimate. We proved the solution exists and is ambiguous. We eventually came up with a variational smooth in the interior of the domain while only 퐶1,1 up approach to tackle this problem. to the boundary. Nirenberg made a joke that we are like Nirenberg-Pogorelov’s isometric embedding theorem “half defeated generals.” But for me, the solution is good allowed us to identify the extra degree of freedom as the enough to solve our geometric problem on affine spheres. time function. Returning to the derivation of Brown-York, After about eight years, Nirenberg finished the strong we discovered that there is indeed a canonical choice of boundary estimate with Luis Caffarelli and . I gauge with respect to each time function, to which a quasi- learned a lot from him in the process of those ten years, local energy (instead of mass) can be assigned. The choice and he has always been encouraging. of this gauge is justified by the positivity of the quasi-local energy, whose proof comprises ideas from Schoen-Yau, References Shi-Tam, Bartnik, Witten, and Liu-Yau. We later realized [1] F. Labourie, Immersions isometriques elliptiques et courbes the choice is closely related to a gravitational conservation pseudo-holomorphes. (French) [Elliptic isometric immersions law which plays an important role in the study of the and pseudoholomorphic curves], J. Differential Geom. 30 dynamics of the Einstein equation. (1989), no. 2, 395–424.

10 Notices of the AMS Volume 63, Number 2 [2] C.-C. M. Liu and S.-T. Yau, Positivity of quasi-local mass II, J. Amer. Math. Soc. 19 (2006), no. 1, 181–204. [3] Louis Nirenberg, The Weyl and Minkowski problems in dif- ferential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. [4] Y. Shi and L.-F. Tam, Positive mass theorem and the bound- ary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. [5] M.-T. Wang and S.-T. Yau, Quasi-local mass in general relativity, Phys. Rev. Lett. 102 (2009), no. 2, no. 021101. [6] , Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys. 288 (2009), no. 3, 919–942.

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