Recent Applications of Nirenberg's Classical Ideas
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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 symmetry 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, New York 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, Louis Nirenberg and John F. Nash Jr. were 0}: awarded the Abel Prize “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 geometric analysis.” While many an article has been written Idescribing Nirenberg’s greatest works, here we have asked mathematicians 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 symmetries 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 mathematics 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: Yakov Sinai Σλ 2013: Pierre Deligne x1 2012: Endre Szemerédi 2011: John Milnor 2010: John Tate 2009: Mikhail Leonidovich Gromov Tλ 2008: John G. Thompson and Jacques Tits 2007: S. R. Srinivasa Varadhan 2006: Lennart Carleson Figure 2. In the Alexandrov method, one reflects 2005: Peter Lax domains Σ휆 across vertical hyperplanes 푇휆 that move continuously from the right to the middle. 2004: Michael Atiyah and Isadore Singer 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 Simon Donaldson, 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 potential theory, 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.