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A Few of Louis Nirenberg’s Many Contributions to the Theory of Partial Differential Equations

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The Abel Prize 2013-2017

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Abstract

Mathematics is the language of science, and partial differential equations are a crucial component: they provide the language we use to describe—and the tools we use to understand—phenomena in many areas including geometry, engineering, and physics.

The author gratefully acknowledges support from NSF through grant DMS-1311833.

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Notes

  1. 1.

    Louis Nirenberg receives the National Medal of Science, Notices Amer. Math. Soc. 43(10), 1111–1116 (1996) (includes “Nirenberg’s work in partial differential equations” by L. Caffarelli, and “Nirenberg’s work in complex analysis” by J. J. Kohn).

  2. 2.

    Donaldson, S.: On the work of Louis Nirenberg. Notices Amer. Math. Soc. 58(3), 469–472 (2011).

  3. 3.

    Interview with Louis Nirenberg, interviewed by A. Jackson. Notices Amer. Math. Soc. 49(4), 441–449 (2002), and Interview with Louis Nirenberg, interviewed by M. Raussen and C. Skau. Newsletter of the European Mathematical Society, Dec 2015, 33–38; reprinted in Notices Amer. Math. Soc. 63(2), 135–140 (2016).

  4. 4.

    Louis Nirenberg, interviewed by Jalal Shatah, on the Simons Foundation’s Science Lives website https://www.simonsfoundation.org/2014/04/21/louis-nirenberg/. (Accessed 8 March 2018).

References

  1. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959).

    Article  MathSciNet  Google Scholar 

  2. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17, 35–92 (1964).

    Article  MathSciNet  Google Scholar 

  3. Agmon, S., Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach space. Comm. Pure Appl. Math. 16, 121–239 (1963).

    Article  MathSciNet  Google Scholar 

  4. Aubin, Th.: Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures et Appl. 55, 269–293 (1976).

    MATH  Google Scholar 

  5. Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain. Comm. Pure Appl. Math. 41, 253–294 (1988).

    Article  MathSciNet  Google Scholar 

  6. Berestycki, H., Caffarelli, L., Nirenberg, L.: Monotonicity for elliptic equations in unbounded Lipschitz domains. Comm. Pure Appl. Math. 50, 1089–1111 (1997).

    Article  MathSciNet  Google Scholar 

  7. Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 69–94 (1997).

    MathSciNet  MATH  Google Scholar 

  8. Berestycki, H., Nirenberg, L.: Monotonicity, symmetry, and antisymmetry of solutions of semilinear elliptic equations. J. Geom. Phys. 5, 237–275 (1988).

    Article  MathSciNet  Google Scholar 

  9. Berestycki, H., Nirenberg, L.: Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, In: Rabinowitz, P., Zehnder, E. (Eds.), Analysis, Et Cetera, pp. 115–164, Academic Press (1990).

    Google Scholar 

  10. Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. (N.S.) 22 1–37 (1991).

    Article  MathSciNet  Google Scholar 

  11. Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Nonlin. 9, 497–572 (1992).

    Article  MathSciNet  Google Scholar 

  12. Brezis, H.: Elliptic equations with limiting Sobolev exponents—the impact of topology. Comm. Pure Appl. Math. 39, S17–S39 (1986).

    Article  MathSciNet  Google Scholar 

  13. Brezis, H.: Symmetry in nonlinear PDE’s. In: Giaquinta, M., Shatah, J., Varadhan, S.R.S. (eds.) Differential Equations—La Pietra 1996. AMS Proc. Symp. Pure Math. 65, 1–12 (1999).

    Google Scholar 

  14. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical exponents. Comm. Pure Appl. Math. 36, 437–477 (1983).

    Article  MathSciNet  Google Scholar 

  15. Browder, F.E.: The Dirichlet problem for linear elliptic equations of arbitrary even order with variable coefficients. Proc. Natl. Acad. Sci. U.S.A. 38, 230–235 (1952).

    Article  MathSciNet  Google Scholar 

  16. Browder, F.E.: Estimates and existence theorems for elliptic boundary value problems: Proc. Natl. Acad. Sci. U.S.A. 45, 365–372 (1959).

    Article  MathSciNet  Google Scholar 

  17. Buckmaster, T., Vicol, V.: Nonuniqueness of weak solutions to the Navier–Stokes equation. Ann. of Math., in press.

    Google Scholar 

  18. Caccioppoli, R.: Ovaloidi di metrica assegnata. Pontificia Academia Scientiarum. Commentationes. 4(1) 1–20 (1940).

    MathSciNet  MATH  Google Scholar 

  19. Caffarelli, L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano 58. 253–284 (1988).

    Article  MathSciNet  Google Scholar 

  20. Caffarelli, L., Kohn, R.V., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982).

    Article  MathSciNet  Google Scholar 

  21. Caffarelli, L., Kohn, R.V., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984).

    MathSciNet  MATH  Google Scholar 

  22. Calderón, A.P., Zygmund, A.: On the existence of singular integrals. Acta. Math. 88, 85–139 (1952).

    Article  MathSciNet  Google Scholar 

  23. Chen, C.-C., Strain, R.M., Tsai, T.-P., Yau, H.-T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations II. Comm. Partial Diff. Eqns. 34, 203–232 (2009).

    Article  MathSciNet  Google Scholar 

  24. Courant R., Hilbert, D.: Methoden der Mathematische Physik, Vol. 2. Springer, Berlin (1937).

    Book  Google Scholar 

  25. De Giorgi, E.: Sulla differenziabilità e lanaliticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3), 3, 25–43 (1957).

    MathSciNet  MATH  Google Scholar 

  26. Douglis, A., Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8, 503–538 (1955).

    Article  MathSciNet  Google Scholar 

  27. Dolbeault, J., Esteban, M.J., Loss, M.: Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces. Invent. Math. 206, 397–440 (2016).

    Article  MathSciNet  Google Scholar 

  28. Druet, O.: Elliptic equations with critical Sobolev exponents in dimension 3. Ann. Inst. H. Poincaré Anal. Nonlin. 19, 125–142 (2002).

    Article  MathSciNet  Google Scholar 

  29. Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77, 587–588 (1971).

    Article  MathSciNet  Google Scholar 

  30. Fefferman, C., Stein, E.: H p spaces of several variables. Acta Math. 129, 137–193 (1972).

    Article  MathSciNet  Google Scholar 

  31. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55, 1461–1506 (2002).

    Article  MathSciNet  Google Scholar 

  32. Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Rational Mech. Anal. 180 (2006) 183–236.

    Article  MathSciNet  Google Scholar 

  33. Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958).

    MathSciNet  MATH  Google Scholar 

  34. Gårding, L.: Dirichlet’s problem for linear elliptic partial differential equations. Math. Scandinavica 1, 55–72 (1953).

    Article  MathSciNet  Google Scholar 

  35. Ghoussoub, N., Robert, F.: Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions. Bull. Math. Sci. 6, 89–144 (2016).

    Article  MathSciNet  Google Scholar 

  36. Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979).

    Article  MathSciNet  Google Scholar 

  37. Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb {R}^n\). In: Mathematical Analysis and Applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7A, Academic Press (1981).

    Google Scholar 

  38. Guillod, J., Šverák, V.: Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces. arXiv:1704.00560 (2017).

    Google Scholar 

  39. Hopf, H.: Lectures on Differential Geometry in the Large, Stanford University (1956).

    Google Scholar 

  40. Jia, H., Šverák, V.: Are the incompressible 3D Navier–Stokes equations locally ill-posed in the natural energy space? J. Funct. Anal. 268, 3734–3766 (2015).

    Article  MathSciNet  Google Scholar 

  41. John, F.: Rotation and strain. Comm. Pure Appl. Math. 14. 391–413 (1961).

    Article  MathSciNet  Google Scholar 

  42. John, F.: Uniqueness of non-linear elastic equilibrium for prescribed boundary displacements and sufficiently small strains. Comm. Pure Appl. Math. 25, 617–634 (1972).

    Article  MathSciNet  Google Scholar 

  43. John. F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426, (1961).

    Article  MathSciNet  Google Scholar 

  44. Koch, G., Nadirashvili, N., Seregin, G.A., Šverák, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203, 83–105 (2009).

    Article  MathSciNet  Google Scholar 

  45. Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18, 269–305 (1965).

    Article  MathSciNet  Google Scholar 

  46. Kukavica, I., Pei, Y.: An estimate on the parabolic fractal dimension of the singular set for solutions of the Navier–Stokes system. Nonlinearity 25, 2775–2783 (2012).

    Article  MathSciNet  Google Scholar 

  47. Ladyzhenskaya, O.A., Seregin, G.A.: On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999).

    Article  MathSciNet  Google Scholar 

  48. Lewy, H.: On the existence of a closed convex surface realizing a given Riemannian metric. Proc. Natl. Acad. Sci. U.S.A. 24, 104–106 (1938).

    Article  Google Scholar 

  49. Lewy, H.: On differential geometry in the large. I. Minkowski’s problem. Trans. Amer. Math. Soc. 43, 258–270 (1938).

    MathSciNet  MATH  Google Scholar 

  50. Li, YanYan: The work of Louis Nirenberg. In: Proceedings of the International Congress of Mathematicians 2010,vol. I, 127–137. Hindustan Book Agency, New Delhi (2010). Also available at http://www.wias-berlin.de/imu/archive/ICM2010/www.icm2010.in/wp-content/icmfiles/laudaions/chern.pdf. (Accessed 8 March 2018.)

  51. Lin, C.-S.: Interpolation inequalities with weights. Comm. Partial Diff. Eqns. 11, 1515–1538 (1986).

    Article  MathSciNet  Google Scholar 

  52. Lin, C.-S., Wang, Z.-Q.: Symmetry of extremal functions for the Caffarelli–Kohn–Nirenberg inequalities. Proc. Amer. Math. Soc. 132, 1686–1691 (2004).

    MATH  Google Scholar 

  53. Lin, F.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51, 241–257 (1998).

    Article  MathSciNet  Google Scholar 

  54. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I and Part II. Ann. Inst. H. Poincaré Anal. Nonlin. 1, 109–145 and 223–283 (1984).

    Google Scholar 

  55. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. Part I and Part II. Rev. Mat. Iberoamericana 1(1), 145–201 and 1(2), 45–121 (1985).

    Google Scholar 

  56. Miranda, C.: Su un problema di Minkowski. Rend. Sem. Mat. Roma 3, 96–108 (1939).

    MathSciNet  MATH  Google Scholar 

  57. Morrey, C.B.: On the solutions of quasilinear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938).

    Article  MathSciNet  Google Scholar 

  58. Moser, J.: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13, 457–468 (1960).

    Article  MathSciNet  Google Scholar 

  59. Moser, J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961).

    Article  MathSciNet  Google Scholar 

  60. Nadirashvili, N., Tkachev, V., Vlăduţ, S.: A non-classical solution to a Hessian equation from Cartan isoparametric cubic. Adv. Math. 231, 1589–1597 (2012).

    Article  MathSciNet  Google Scholar 

  61. Nadirashvili, N., Tkachev, V., Vlăduţ, S.: Nonlinear Elliptic Equations and Nonassociative Algebras. Mathematical Surveys & Monographs 200, American Mathematical Society (2014).

    Google Scholar 

  62. Nash, J.: Parabolic equations. Proc. Natl. Acad. Sci. U.S.A. 43. 754–758 (1957).

    Article  MathSciNet  Google Scholar 

  63. Nečas, J., Rŭžička, M., Šverák, V.: On Leray’s self-similar solutions of the Navier–Stokes equations. Acta Math. 176, 283–294 (1996).

    Article  MathSciNet  Google Scholar 

  64. Nguyen, H.-M., Squassina, M.: Fractional Caffarelli–Kohn–Nirenberg inequalities. J. Funct. Anal. 274(9), 2661–2672 (2018).

    Article  MathSciNet  Google Scholar 

  65. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. of Math. 65, 391–404 (1957).

    Article  MathSciNet  Google Scholar 

  66. Nirenberg, L.: The Determination of a Closed Convex Surface Having Given Line Element. PhD Thesis, New York University (1949).

    Google Scholar 

  67. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6, 337–394 (1953).

    Article  MathSciNet  Google Scholar 

  68. Nirenberg, L.: On nonlinear elliptic partial differential equations and Hölder continuity. Comm. Pure Appl. Math. 6, 103–156 (1953).

    Article  MathSciNet  Google Scholar 

  69. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13, 115–162 (1959).

    MathSciNet  MATH  Google Scholar 

  70. Nirenberg, L.: Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. (N.S.) 4, 267–302 (1981).

    Article  MathSciNet  Google Scholar 

  71. Nirenberg, L.: Topics in Nonlinear Functional Analysis, Courant Lecture Notes Vol. 6, Courant Institute of Mathematical Sciences, New York, and American Mathematical Society, Providence (2001).

    Google Scholar 

  72. Pazy, A.: Asymptotic expansions of solutions of ordinary differential equations in Hilbert space. Arch. Rational Mech. Anal. 24, 193–218 (1967).

    Article  MathSciNet  Google Scholar 

  73. Rivière, T.: Exploring the unknown: the work of Louis Nirenberg on partial differential equations. Notices Amer. Math. Soc. 63(2), 120–125 (2016). A longer, more detailed version is available as arXiv:1505.04930.

    Article  MathSciNet  Google Scholar 

  74. Robinson, J.C., Rodrigo, J.L., Sadowski, W.: The Three-Dimensional Navier–Stokes Equations: Classical Theory. Cambridge University Press (2016).

    Google Scholar 

  75. Scheffer, V.: Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys. 55, 97–112 (1977).

    Article  MathSciNet  Google Scholar 

  76. Scheffer, V.: The Navier–Stokes equations on a bounded domain. Comm. Math. Phys. 73, 1–42 (1980).

    Article  MathSciNet  Google Scholar 

  77. Scheffer, V.: Nearly one-dimensional singularities of solutions to the Navier–Stokes inequality. Comm. Math. Phys. 110, 525–551 (1987).

    Article  MathSciNet  Google Scholar 

  78. Seregin, G., Šverák, V.: On type I singularities of the local axisymmetric solutions of the Navier–Stokes equations. Comm. Partial Diff. Eqns. 34, 171–201 (2009).

    Article  Google Scholar 

  79. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971).

    Article  MathSciNet  Google Scholar 

  80. Sormani, C.: Recent applications of Nirenberg’s classical ideas. Notices Amer. Math. Soc. 63(2), 126–134 (2016) (includes “The Gidas–Ni–Nirenberg theorem” by X. Cabré; “Nirenberg’s problem” by S.-Y. A. Chang; “Caffarelli–Kohn–Nirenberg and the Navier–Stokes problem” by G. Seregin; “Stability of the Gagliardo–Nirenberg Sobolev inequality” by E. Carlen and A. Figalli; and “Isometric embeddings of surfaces” by M.-T. Wang and S.-T. Yau).

    Google Scholar 

  81. Stoker, J.J.: On the uniqueness theorems for the embedding of convex surfaces in three-dimensional space. Comm. Pure Appl. Math. 3, 231–257 (1950).

    Article  MathSciNet  Google Scholar 

  82. Tsai, T.-P.: On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal. 143, 29–51 (1998).

    Article  MathSciNet  Google Scholar 

  83. Vishik, M.I.: On strongly elliptic systems of differential equations. Mat. Sbornik, N.S. 29(71), 615–676 (1951).

    MathSciNet  Google Scholar 

  84. Weyl, H.: Über die bestimmung einer geschlossenen konvexen fläche durch ihr linienelement. Vierteljahrsschrift der Naturforschenden Gessellschaft, Zürich 61, 40–72 (1916).

    Google Scholar 

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  1. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Robert V. Kohn

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    Helge Holden

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Ragni Piene

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Kohn, R.V. (2019). A Few of Louis Nirenberg’s Many Contributions to the Theory of Partial Differential Equations. In: Holden, H., Piene, R. (eds) The Abel Prize 2013-2017. The Abel Prize. Springer, Cham. https://doi.org/10.1007/978-3-319-99028-6_20

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