Skip to main content Accessibility help


  • Nikolay Kuznetsov (a1) and Alexander Nazarov (a2) (a3)


During the past 55 years substantial progress concerning sharp constants in Poincaré-type and Steklov-type inequalities has been achieved. Original results of H. Poincaré, V. A. Steklov and his disciples are reviewed along with the main further developments in this area.



Hide All
1.Acosta, G. and Durán, R., An optimal Poincaré inequality in L 1 for convex domains. Proc. Amer. Math. Soc. 132 2004, 195202.
2.Adimurthiand Mancini, G., The Neumann problem for elliptic equations with critical nonlinearity. In Nonlinear Analysis, Quaderni della Scuola Normale Superiore (Pisa, 1991), 925.
3.Agarwal, R. P., Bohner, M., O’Regan, D. and Saker, D. H., Some dynamic Wirtinger-type inequalities and their applications. Pacific J. Math. 252 2011, 118.
4.Almansi, E., Sorpa una delle esperienze di Plateau. Ann. Mat. Pura Appl. (3) 12 1905, 117.
5.Arnold, V. I., On teaching mathematics. Uspekhi Mat. Nauk. 53(1) 1998, 229234 (in Russian); Engl. transl. Russian Math. Surveys 53 (1998), 229–236.
6.Ashbaugh, M. S. and Benguria, R. D., Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions. SIAM J. Math. Anal. 24 1993, 557570.
7.Aubin, T., Problèmes isopérimetriques et espaces de Sobolev. J. Differential Geom. 11 1976, 573598. See also T. Aubin, C. R. Acad. Sci. Paris 280 (1975), 279–281.
8.Barbosa, L. and Bérard, P., Eigenvalue and “twisted” eigenvalue problems, applications to CMC surfaces. J. Math. Pures Appl. 79 2000, 427450.
9.Bebendorf, M., A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4) 2003, 751756.
10.Beckenbach, E. F. and Bellman, R., Inequalities, Springer (Berlin, 1961).
11.Bennewitz, C. and Saitō, Y., An embedding norm and the Lindqvist trigonometric functions. Electron. J. Differential Equations 2002 2002, 16.
12.Bennewitz, C. and Saitō, Y., Approximation numbers of Sobolev embedding operators on an interval. J. Lond. Math. Soc. (2) 70 2004, 244260.
13.Blaschke, W., Kreis und Kugel, Verlag von Veit (Leipzig, 1916).
14.Bliss, G. A., An integral inequality. J. Lond. Math. Soc. (2) 5 1930, 4046.
15.Bojarski, B., Remarks on Sobolev imbedding inequalities. In Proc. Conf. Complex Analysis, Joensuu, 1987 (Lecture Notes in Mathematics 1351), Springer (Berlin, 1989), 5268.
16.Boyd, D. W., Best constants in a class of integral inequalities. Pacific J. Math. 30 1969, 367383.
17.Brandolini, B., Della Pietra, F., Nitsch, C. and Trombetti, C., Symmetry breaking in a constrained Cheeger type isoperimetric inequality. Preprint, 7 November 2013, arXiv:1305.6271v2 [math.OC].
18.Brock, F., An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81 2001, 6971.
19.Buslaev, A. P., Kondrat’ev, V. A. and Nazarov, A. I., On a family of extremal problems and related properties of an integral. Mat. Zametki 64 1998, 830838 (in Russian); Engl. transl. Math. Notes 64 (1999) 719–725.
20.Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (papers dedicated to Salomon Bochner, 1969), Princeton University Press (Princeton, NJ, 1970), 195199.
21.Cianchi, A., A sharp form of Poincaré inequalities on balls and spheres. Z. Angew. Math. Phys. 40 1989, 558569.
22.Cianchi, A., A sharp trace inequality for functions of bounded variation in the ball. Proc. Roy. Soc. Edinburgh A 142 2012, 11791191.
23.Cianchi, A., Ferone, V., Nitsch, C. and Trombetti, C., Balls minimize trace constants in $BV$. Preprint, 24 January 2013, arXiv:1301.5770 [math.OC].
24.Cordero-Erausquin, D., Nazaret, B. and Villani, C., A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182 2004, 307332.
25.Courant, R. and Hilbert, D., Methoden der mathematischen Physik. II, Springer (Berlin, 1937).
26.Croce, G., Henrot, A. and Pisante, G., An isoperimetric inequality for a nonlinear eigenvalue problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 2012, 2134.
27.Dacorogna, B., Gangbo, W. and Subia, N., Sur une généralisation de l’inégalité de Wirtinger. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 1992, 2950.
28.Demyanov, A. V. and Nazarov, A. I., On the existence of an extremal function in Sobolev embedding theorems with critical exponents. Algebra i Analiz 17(5) 2005, 105140 (in Russian); Engl. transl. St. Petersburg Math. J. 17(5) (2006), 108–142.
29.Deny, J. and Lions, J.-L., Les espace du type de Beppo Levi. Ann. Inst. Fourier 5 1953–1954, 305370.
30.Egorov, Yu. V. and Kondratiev, V. A., On a Lagrange problem. C. R. Acad. Sci. Paris Ser. I 317 1993, 903908.
31.Escobar, J. F., Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37 1988, 687698.
32.Esposito, L., Nitsch, C. and Trombetti, C., Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20 2013, 253264.
33.Faber, G., Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitzungsber. Bayer. Akad. Wiss. Math.-Phys. Kl. 1923, 169172.
34.Fan, K., Taussky, O. and Todd, J., Discrete analogs of inequalities of Wirtinger. Monatsh. Math. 59 1955, 7390.
35.Federer, H. and Fleming, W. H., Normal and integral currents. Ann. Math. 72 1960, 458520.
36.Ferone, V., Nitsch, C. and Trombetti, C., A remark on optimal weighted Poincaré inequalities for convex domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23(4) 2012, 467475.
37.Filonov, N. D., On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator. Algebra i Analiz 16(2) 2004, 172176 (in Russian); Engl. transl. St. Petersburg Math. J. 16 (2005), 413–416.
38.Freitas, P., Upper and lower bounds for the first Dirichlet eigenvalue of a triangle. Proc. Amer. Math. Soc. 134 2006, 20832089.
39.Freitas, P. and Henrot, A., On the first twisted Dirichlet eigenvalue. Comm. Anal. Geom. 12(5) 2004, 10831103.
40.Friedrichs, K.-O., Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzberganges vom Einsteinschen zum Newtonschen Gesetz. Math. Ann. 98 1927, 566575.
41.Gagliardo, E., Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in più variabili. Rend. Sem. Mat. Univ. Padova 27 1957, 284305.
42.Gerasimov, I. V. and Nazarov, A. I., Best constant in a three-parameter Poincaré inequality. Probl. Mat. Anal. 61 2011, 6986 (in Russian); Engl. transl. J. Math. Sci. 179 (2011), 80–99.
43.Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, Cambridge University Press (Cambridge, 1934).
44.Hille, E., Jacob David Tamarkin — his life and work. Bull. Amer. Math. Soc. 53 1947, 440457.
45.Jaye, B. J., Maz’ya, V. G. and Verbitsky, I. E., Existence and regularity of positive solutions of elliptic equations of Schrödinger type. J. Anal. Math. 118 2012, 577621.
46.Jaye, B. J., Maz’ya, V. G. and Verbitsky, I. E., Quasilinear elliptic equations and weighted Sobolev–Poincaré inequalities with distributional weights. Adv. Math. 232 2013, 513542.
47.John, F., Rotation and strain. Comm. Pure Appl. Math. 14 1961, 391413.
48.Kawohl, B. and Fridman, V., Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin. 44 2003, 659667.
49.Kneser, A., Wladimir Stekloff zum Gedächtnis. Jahresber. Dtsch. Math.-Ver. 38 1929, 206231.
50.Krahn, E., Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94 1925, 97100.
51.Krahn, E., Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Comment. Univ. Tartu (Dorpat) A9 1926, 1144.
52.Kresin, G. and Maz’ya, V., Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, American Mathematical Society (Providence, RI, 2012).
53.Kriloff (N. M. Krylov), N. M., Sur certaines inégalités trouvées dans l’exposition de la méthode de Schwarz–Poincaré–Stekloff et quon rencontre dans la résolution de nombreax problèmes de minimum. Notes Inst. Mines 6(1) 1915, 1014 (in Russian); French resumé. See also N. M. Krylov, Selected Papers, Vol. 1, Academy of Sciences of the Ukrainian SSR (Kiev, 1960), 113–120.
54.Kuznetsov, N., The legacy of Vladimir Andreevich Steklov in mathematical physics: work and school. EMS Newslett.(91) 2014, 3138.
55.Kuznetsov, N., Kulczycki, T., Kwaśnicki, M., Nazarov, A., Poborchi, S., Polterovich, I. and Siudeja, B., The legacy of Vladimir Andreevich Steklov. Notices Amer. Math. Soc. 61 2014, 922.
56.Lamb, H., Hydrodynamics, Cambridge University Press (Cambridge, 1932).
57.Laugesen, R. S. and Siudeja, B. A., Maximizing Neumann fundamental tones of triangles. J. Math. Phys. 50 2009, 112903 doi:10.1063/1.3246834.
58.Lefton, L. and Wei, D., Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method. Numer. Funct. Anal. Optim. 18 1997, 389399.
59.Levin, V. I., Notes on inequalities. II. On a class of integral inequalities. Mat. Sb. 4 1938, 309324 (in Russian).
60.Lindqvist, P., Some remarkable sine and cosine functions. Ric. Mat. 44 1995, 269290.
61.Lyapunov, A. M., Investigation of a special case in the problem of stability of motion. Mat. Sb. 17 1893, 253333 (in Russian); see also A. M. Lyapunov, Collected Papers, Vol. 2, Academy of Sciences of the SSSR (Moscow, 1956), 277–336.
62.Maz’ya, V. G., Classes of domains and imbedding theorems for function spaces. Dokl. AN SSSR 133 1960, 527530 (in Russian); Engl. transl. Soviet Math. Dokl. 133 (1960), 882–885.
63.Maz’ya, V. G., Classes of domains, measures and capacities in the theory of differentiable functions. In Analiz–3 (Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Fundamental’nye Napravleniya 26), VINITI (Moscow, 1988), 159228 (in Russian); Engl. transl. In Analysis, III (Encyclopaedia of Mathematical Sciences 26), Springer (Berlin, 1991), 141–211.
64.Maz’ya, V. G., Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer (Heidelberg, 2011).
65.Mikhlin, S. G., Konstanten in einigen Ungleichungen der Analysis (Teubner-Texte zur Matematik 35), Teubner (Leipzig, 1981) ; Engl. transl. Constants in Some Inequalities of Analysis, Wiley-Interscience (Chichester, 1986).
66.Mitrea, D. and Mitrea, M., On the scientific work of V. G. Maz’ya: a personalized account. In Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday (Proceedings of Symposia in Pure Mathematics 79), American Mathematical Society (Providence, RI, 2008), viixvii.
67.Mitrinović, D. S., Pečarić, J. E. and Fink, A. M., Inequalities Involving Functions and their Integrals and Derivatives, Kluwer (Dordrecht, 1991).
68.Mitrinović, D. S. and Vasić, P. M., An integral inequality ascribed to Wirtinger, and its variations and generalizations. Publ. Fac. Electrotech. Univ. Belgrade Sér. Math. Phys.(272) 1969, 157170.
69.Nazaret, B., Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65 2006, 19771985.
70.Nazarov, A. I., On exact constant in the generalized Poincaré inequality. Probl. Mat. Anal. 24 2002, 155180 (in Russian); Engl. transl. J. Math. Sci. 112 (2002), 4029–4047.
71.Nazarov, A. I., Dirichlet and Neumann problems to critical Emden–Fowler type equations. J. Global Optim. 40 2008, 289303.
72.Nazarov, A. I., Trace Hardy–Sobolev inequalities in cones. Algebra i Analiz 22(6) 2010, 200213 (in Russian); Engl. transl. St. Petersburg Math. J. 22 (2011), 997–1006.
73.Nazarov, A. I., On symmetry and asymmetry in a problem of shape optimization. Preprint, 17 August 2012, arXiv:1208.3640 [math.AP].
74.Nazarov, A. I. and Poborchi, S. V., The Poincaré Inequality and its Applications, St. Petersburg University Press (St. Petersburg, 2012) (in Russian).
75.Nazarov, A. I. and Poborchi, S. V., On validity conditions for the Poincaré inequality. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 410 2013, 104109 (in Russian); Engl. transl. J. Math. Sci. 195 (2013), 61–63.
76.Nazarov, A. I. and Repin, S. I., Exact constants in Poincaré-type inequalities for functions with zero boundary traces. Preprint, 9 November 2012, arXiv:1211.2224 [math.AP].
77.Nazarov, A. I. and Reznikov, A. B., On the existence of an extremal function in critical Sobolev trace embedding theorem. J. Funct. Anal. 258 2010, 39063921.
78.Nikitin, Ya., Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press (1995).
79.Nikodým, O., Sur une classe de fonctions considérées dans l’étude du problème de Dirichlet. Fund. Math. 21 1933, 129150.
80.Parini, E., An introduction to the Cheeger problem. Surv. Math. Appl. 6 2011, 921.
81.Payne, L. E. and Weinberger, H. F., An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 1960, 286292.
82.Peetre, J., MR 655789 (84a:46076). Review of S. G. Michlin, Konstanten in einigen Ungleichungen der Analysis (Teubner-Texte zur Mathematik 35), Teubner (Leipzig, 1981).
83.Penskoi, A. V., Metrics extremal for eigenvalues of Laplace–Beltrami operator on surfaces. Uspekhi Mat. Nauk 68(6) 2013, 107168 (in Russian); Engl. transl. Russian Math. Surveys 68 (2013), 1073–1130.
84.Poincaré, H., Sur les équations aux dériveés partielles de la physique mathematique. Amer. J. Math. 12 1890, 211294.
85.Poincaré, H., Sur les équations de la physique mathematique. Rend. Circ. Mat. Palermo 8 1894, 57155.
86.Pólya, G. and Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Princeton University Press (Princeton, NJ, 1951).
87.Rellich, F., Darstellung der Eigenwerte von Δu +𝜆 u = 0 durch ein Randintegral. Math. Z. 46 1940, 635636.
88.Repin, S. I., A Posteriori Error Estimates for Partial Differential Equations, Walter de Gruyter (Berlin, 2008).
89.Rosen, G., Minimum value for c in the Sobolev inequality ∥𝜑3∥⩽c∥𝜑∥3. SIAM J. Appl. Math. 21 1971, 3032.
90.Scheeffer, L., Über die Bedeutung der Begriffe “Maximum und Minimum” in der Variationsrechung. Math. Ann. 26 1885, 197208.
91.Schmidt, E., Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet. Math. Ann. 117 1940, 301326.
92.Sobolev, S. L., On a theorem of functional analysis. Mat. Sb. (N.S.) 4 1938, 471497 (in Russian); Engl. transl. Transl. Amer. Math. Soc. Ser. 2 34 (1963), 39–68.
93.Stanoyevitch, A., Products of Poincaré domains. Proc. Amer. Math. Soc. 117 1993, 7987.
94.Steinerberger, S., Sharp $L^{1}$-Poincaré inequalities correspond to optimal hypersurface cuts, Preprint, 15 November 2013, arXiv:1309.6211v3 [math.CA].
95.Steklov, V. A., On the expansion of a given function into a series of harmonic functions. Commun. Kharkov Math. Soc. Ser. 2 5 1896, 6073 (in Russian).
96.Steklov, V. A., The problem of cooling of an heterogeneous rigid rod. Commun. Kharkov Math. Soc. Ser. 2 5 1896, 136181 (in Russian).
97.Steklov, V. A., On the expansion of a given function into a series of harmonic functions. Commun. Kharkov Math. Soc. Ser. 2 6 1897, 57124 (in Russian).
98.Steklov, V. A., Osnovnye Zadachi Matematicheskoy Fiziki (Fundamental Problems of Mathematical Physics 1), Russian Academy of Sciences (Petrograd, 1922) (in Russian).
99.Stekloff (V. A. Steklov), W., Problème de refroidissement d’une barre hétérogène. Ann. Fac. Sci. Toulouse Sér. 2 3 1901, 281313.
100.Stekloff (V. A. Steklov), W., Sur certaines égalités remarquables. C. R. Acad. Sci. Paris 135 1902, 783786.
101.Stekloff (V. A. Steklov), W., Sur les problèmes fondamentaux de la physique mathematique (suite et fin). Ann. Sci. ENS Sér. 3 19 1902, 455490.
102.Stekloff (V. A. Steklov), W., Sur la condition de fermeture des systèmes de fonctions orthogonales. C. R. Acad. Sci. Paris 151 1910, 11161119.
103.Stekloff (V. A. Steklov), W., Sur la théorie de fermeture des systèmes de fonctions orthogonales dépendant d’un nombre quelconque des variables. Mém. Acad. Sci. St. Pétersbourg Cl. Phys. Math. Sér. 8 30(4) 1911, 187.
104.Strutt (Lord Rayleigh), J. W., The Theory of Sound, 2nd edn, Vo1. 1, Macmillan (London, 1894).
105.Szegő, G., Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3 1954, 343356.
106.Talenti, G., Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 1976, 353372.
107.Tamarkine (J. D. Tamarkin), Ja. D., Application de la méthode des fonctions fondamentales à l’étude de l’équation différentielle des verges vibrantes élastiques. Commun. Kharkov Math. Soc. Ser. 2 12 1910, 1946.
108.Valtorta, D., Sharp estimate on the first eigenvalue of the p-Laplacian. Nonlinear Anal. 75 2012, 49744994.
109.Wang, X. J., Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93(2) 1991, 283310.
110.Weinberger, H. F., An isoperimetric inequality for the n-dimensional free membrane problem. J. Ration. Mech. Anal. 5 1956, 633636.
111.Weinstock, R., Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3 1954, 745753.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed