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SHARP CONSTANTS IN THE POINCARÉ, STEKLOV AND RELATED INEQUALITIES (A SURVEY)

Part of: Linear function spaces and their duals Partial differential equations Inequalities

Published online by Cambridge University Press:  10 October 2014

Nikolay Kuznetsov  and
Alexander Nazarov
Nikolay Kuznetsov
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Science, St. Petersburg, Russia email [email protected]
Alexander Nazarov
Affiliation:
Laboratory of Mathematical Physics, St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia email [email protected]
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Abstract

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.

MSC classification

Primary: 26D10: Inequalities involving derivatives and differential and integral operators 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 328 - 344
Copyright
Copyright © University College London 2014 

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References

Acosta, G. and Durán, R., An optimal Poincaré inequality in L 1 for convex domains. Proc. Amer. Math. Soc. 132 2004, 195–202.CrossRefGoogle Scholar
Adimurthiand Mancini, G., The Neumann problem for elliptic equations with critical nonlinearity. In Nonlinear Analysis, Quaderni della Scuola Normale Superiore (Pisa, 1991), 9–25.Google Scholar
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, 1–18.CrossRefGoogle Scholar
Almansi, E., Sorpa una delle esperienze di Plateau. Ann. Mat. Pura Appl. (3) 12 1905, 1–17.Google Scholar
Arnold, V. I., On teaching mathematics. Uspekhi Mat. Nauk. 53(1) 1998, 229–234 (in Russian); Engl. transl. Russian Math. Surveys 53 (1998), 229–236.Google Scholar
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, 557–570.CrossRefGoogle Scholar
Aubin, T., Problèmes isopérimetriques et espaces de Sobolev. J. Differential Geom. 11 1976, 573–598. See also T. Aubin, C. R. Acad. Sci. Paris 280 (1975), 279–281.CrossRefGoogle Scholar
Barbosa, L. and Bérard, P., Eigenvalue and “twisted” eigenvalue problems, applications to CMC surfaces. J. Math. Pures Appl. 79 2000, 427–450.CrossRefGoogle Scholar
Bebendorf, M., A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4) 2003, 751–756.CrossRefGoogle Scholar
Beckenbach, E. F. and Bellman, R., Inequalities, Springer (Berlin, 1961).CrossRefGoogle Scholar
Bennewitz, C. and Saitō, Y., An embedding norm and the Lindqvist trigonometric functions. Electron. J. Differential Equations 2002 2002, 1–6.Google Scholar
Bennewitz, C. and Saitō, Y., Approximation numbers of Sobolev embedding operators on an interval. J. Lond. Math. Soc. (2) 70 2004, 244–260.CrossRefGoogle Scholar
Blaschke, W., Kreis und Kugel, Verlag von Veit (Leipzig, 1916).CrossRefGoogle Scholar
Bliss, G. A., An integral inequality. J. Lond. Math. Soc. (2) 5 1930, 40–46.CrossRefGoogle Scholar
Bojarski, B., Remarks on Sobolev imbedding inequalities. In Proc. Conf. Complex Analysis, Joensuu, 1987 (Lecture Notes in Mathematics 1351), Springer (Berlin, 1989), 52–68.Google Scholar
Boyd, D. W., Best constants in a class of integral inequalities. Pacific J. Math. 30 1969, 367–383.CrossRefGoogle Scholar
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].CrossRefGoogle Scholar
Brock, F., An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81 2001, 69–71.3.0.CO;2-#>CrossRefGoogle Scholar
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, 830–838 (in Russian); Engl. transl. Math. Notes 64 (1999) 719–725.Google Scholar
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), 195–199.Google Scholar
Cianchi, A., A sharp form of Poincaré inequalities on balls and spheres. Z. Angew. Math. Phys. 40 1989, 558–569.CrossRefGoogle Scholar
Cianchi, A., A sharp trace inequality for functions of bounded variation in the ball. Proc. Roy. Soc. Edinburgh A 142 2012, 1179–1191.CrossRefGoogle Scholar
Cianchi, A., Ferone, V., Nitsch, C. and Trombetti, C., Balls minimize trace constants in $BV$. Preprint, 24 January 2013, arXiv:1301.5770 [math.OC].Google Scholar
Cordero-Erausquin, D., Nazaret, B. and Villani, C., A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182 2004, 307–332.CrossRefGoogle Scholar
Courant, R. and Hilbert, D., Methoden der mathematischen Physik. II, Springer (Berlin, 1937).CrossRefGoogle Scholar
Croce, G., Henrot, A. and Pisante, G., An isoperimetric inequality for a nonlinear eigenvalue problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 2012, 21–34.CrossRefGoogle Scholar
Dacorogna, B., Gangbo, W. and Subia, N., Sur une généralisation de l’inégalité de Wirtinger. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 1992, 29–50.CrossRefGoogle Scholar
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, 105–140 (in Russian); Engl. transl. St. Petersburg Math. J. 17(5) (2006), 108–142.Google Scholar
Deny, J. and Lions, J.-L., Les espace du type de Beppo Levi. Ann. Inst. Fourier 5 1953–1954, 305–370.CrossRefGoogle Scholar
Egorov, Yu. V. and Kondratiev, V. A., On a Lagrange problem. C. R. Acad. Sci. Paris Ser. I 317 1993, 903–908.Google Scholar
Escobar, J. F., Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37 1988, 687–698.CrossRefGoogle Scholar
Esposito, L., Nitsch, C. and Trombetti, C., Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20 2013, 253–264.Google Scholar
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, 169–172.Google Scholar
Fan, K., Taussky, O. and Todd, J., Discrete analogs of inequalities of Wirtinger. Monatsh. Math. 59 1955, 73–90.CrossRefGoogle Scholar
Federer, H. and Fleming, W. H., Normal and integral currents. Ann. Math. 72 1960, 458–520.CrossRefGoogle Scholar
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, 467–475.Google Scholar
Filonov, N. D., On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator. Algebra i Analiz 16(2) 2004, 172–176 (in Russian); Engl. transl. St. Petersburg Math. J. 16 (2005), 413–416.Google Scholar
Freitas, P., Upper and lower bounds for the first Dirichlet eigenvalue of a triangle. Proc. Amer. Math. Soc. 134 2006, 2083–2089.CrossRefGoogle Scholar
Freitas, P. and Henrot, A., On the first twisted Dirichlet eigenvalue. Comm. Anal. Geom. 12(5) 2004, 1083–1103.CrossRefGoogle Scholar
Friedrichs, K.-O., Eine invariante Formulierung des Newtonschen Gravititationsgesetzes und des Grenzberganges vom Einsteinschen zum Newtonschen Gesetz. Math. Ann. 98 1927, 566–575.CrossRefGoogle Scholar
Gagliardo, E., Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in più variabili. Rend. Sem. Mat. Univ. Padova 27 1957, 284–305.Google Scholar
Gerasimov, I. V. and Nazarov, A. I., Best constant in a three-parameter Poincaré inequality. Probl. Mat. Anal. 61 2011, 69–86 (in Russian); Engl. transl. J. Math. Sci. 179 (2011), 80–99.Google Scholar
Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, Cambridge University Press (Cambridge, 1934).Google Scholar
Hille, E., Jacob David Tamarkin — his life and work. Bull. Amer. Math. Soc. 53 1947, 440–457.CrossRefGoogle Scholar
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, 577–621.CrossRefGoogle Scholar
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, 513–542.CrossRefGoogle Scholar
John, F., Rotation and strain. Comm. Pure Appl. Math. 14 1961, 391–413.CrossRefGoogle Scholar
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, 659–667.Google Scholar
Kneser, A., Wladimir Stekloff zum Gedächtnis. Jahresber. Dtsch. Math.-Ver. 38 1929, 206–231.Google Scholar
Krahn, E., Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94 1925, 97–100.CrossRefGoogle Scholar
Krahn, E., Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Comment. Univ. Tartu (Dorpat) A9 1926, 1–144.Google Scholar
Kresin, G. and Maz’ya, V., Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, American Mathematical Society (Providence, RI, 2012).CrossRefGoogle Scholar
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, 10–14 (in Russian); French resumé. See also N. M. Krylov, Selected Papers, Vol. 1, Academy of Sciences of the Ukrainian SSR (Kiev, 1960), 113–120.Google Scholar
Kuznetsov, N., The legacy of Vladimir Andreevich Steklov in mathematical physics: work and school. EMS Newslett.(91) 2014, 31–38.Google Scholar
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, 9–22.CrossRefGoogle Scholar
Lamb, H., Hydrodynamics, Cambridge University Press (Cambridge, 1932).Google Scholar
Laugesen, R. S. and Siudeja, B. A., Maximizing Neumann fundamental tones of triangles. J. Math. Phys. 50 2009, 112903 doi:10.1063/1.3246834.CrossRefGoogle Scholar
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, 389–399.CrossRefGoogle Scholar
Levin, V. I., Notes on inequalities. II. On a class of integral inequalities. Mat. Sb. 4 1938, 309–324 (in Russian).Google Scholar
Lindqvist, P., Some remarkable sine and cosine functions. Ric. Mat. 44 1995, 269–290.Google Scholar
Lyapunov, A. M., Investigation of a special case in the problem of stability of motion. Mat. Sb. 17 1893, 253–333 (in Russian); see also A. M. Lyapunov, Collected Papers, Vol. 2, Academy of Sciences of the SSSR (Moscow, 1956), 277–336.Google Scholar
Maz’ya, V. G., Classes of domains and imbedding theorems for function spaces. Dokl. AN SSSR 133 1960, 527–530 (in Russian); Engl. transl. Soviet Math. Dokl. 133 (1960), 882–885.Google Scholar
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), 159–228 (in Russian); Engl. transl. In Analysis, III (Encyclopaedia of Mathematical Sciences 26), Springer (Berlin, 1991), 141–211.Google Scholar
Maz’ya, V. G., Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer (Heidelberg, 2011).CrossRefGoogle Scholar
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).Google Scholar
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), vii–xvii.CrossRefGoogle Scholar
Mitrinović, D. S., Pečarić, J. E. and Fink, A. M., Inequalities Involving Functions and their Integrals and Derivatives, Kluwer (Dordrecht, 1991).CrossRefGoogle Scholar
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, 157–170.Google Scholar
Nazaret, B., Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65 2006, 1977–1985.CrossRefGoogle Scholar
Nazarov, A. I., On exact constant in the generalized Poincaré inequality. Probl. Mat. Anal. 24 2002, 155–180 (in Russian); Engl. transl. J. Math. Sci. 112 (2002), 4029–4047.Google Scholar
Nazarov, A. I., Dirichlet and Neumann problems to critical Emden–Fowler type equations. J. Global Optim. 40 2008, 289–303.CrossRefGoogle Scholar
Nazarov, A. I., Trace Hardy–Sobolev inequalities in cones. Algebra i Analiz 22(6) 2010, 200–213 (in Russian); Engl. transl. St. Petersburg Math. J. 22 (2011), 997–1006.Google Scholar
Nazarov, A. I., On symmetry and asymmetry in a problem of shape optimization. Preprint, 17 August 2012, arXiv:1208.3640 [math.AP].Google Scholar
Nazarov, A. I. and Poborchi, S. V., The Poincaré Inequality and its Applications, St. Petersburg University Press (St. Petersburg, 2012) (in Russian).Google Scholar
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, 104–109 (in Russian); Engl. transl. J. Math. Sci. 195 (2013), 61–63.Google Scholar
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].Google Scholar
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, 3906–3921.CrossRefGoogle Scholar
Nikitin, Ya., Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press (1995).CrossRefGoogle Scholar
Nikodým, O., Sur une classe de fonctions considérées dans l’étude du problème de Dirichlet. Fund. Math. 21 1933, 129–150.CrossRefGoogle Scholar
Parini, E., An introduction to the Cheeger problem. Surv. Math. Appl. 6 2011, 9–21.Google Scholar
Payne, L. E. and Weinberger, H. F., An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 1960, 286–292.CrossRefGoogle Scholar
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).Google Scholar
Penskoi, A. V., Metrics extremal for eigenvalues of Laplace–Beltrami operator on surfaces. Uspekhi Mat. Nauk 68(6) 2013, 107–168 (in Russian); Engl. transl. Russian Math. Surveys 68 (2013), 1073–1130.Google Scholar
Poincaré, H., Sur les équations aux dériveés partielles de la physique mathematique. Amer. J. Math. 12 1890, 211–294.CrossRefGoogle Scholar
Poincaré, H., Sur les équations de la physique mathematique. Rend. Circ. Mat. Palermo 8 1894, 57–155.CrossRefGoogle Scholar
Pólya, G. and Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Princeton University Press (Princeton, NJ, 1951).Google Scholar
Rellich, F., Darstellung der Eigenwerte von Δu +𝜆 u = 0 durch ein Randintegral. Math. Z. 46 1940, 635–636.CrossRefGoogle Scholar
Repin, S. I., A Posteriori Error Estimates for Partial Differential Equations, Walter de Gruyter (Berlin, 2008).CrossRefGoogle Scholar
Rosen, G., Minimum value for c in the Sobolev inequality ∥𝜑3∥⩽c∥𝜑∥3. SIAM J. Appl. Math. 21 1971, 30–32.CrossRefGoogle Scholar
Scheeffer, L., Über die Bedeutung der Begriffe “Maximum und Minimum” in der Variationsrechung. Math. Ann. 26 1885, 197–208.CrossRefGoogle Scholar
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, 301–326.CrossRefGoogle Scholar
Sobolev, S. L., On a theorem of functional analysis. Mat. Sb. (N.S.) 4 1938, 471–497 (in Russian); Engl. transl. Transl. Amer. Math. Soc. Ser. 2 34 (1963), 39–68.Google Scholar
Stanoyevitch, A., Products of Poincaré domains. Proc. Amer. Math. Soc. 117 1993, 79–87.Google Scholar
Steinerberger, S., Sharp $L^{1}$-Poincaré inequalities correspond to optimal hypersurface cuts, Preprint, 15 November 2013, arXiv:1309.6211v3 [math.CA].Google Scholar
Steklov, V. A., On the expansion of a given function into a series of harmonic functions. Commun. Kharkov Math. Soc. Ser. 2 5 1896, 60–73 (in Russian).Google Scholar
Steklov, V. A., The problem of cooling of an heterogeneous rigid rod. Commun. Kharkov Math. Soc. Ser. 2 5 1896, 136–181 (in Russian).Google Scholar
Steklov, V. A., On the expansion of a given function into a series of harmonic functions. Commun. Kharkov Math. Soc. Ser. 2 6 1897, 57–124 (in Russian).Google Scholar
Steklov, V. A., Osnovnye Zadachi Matematicheskoy Fiziki (Fundamental Problems of Mathematical Physics 1), Russian Academy of Sciences (Petrograd, 1922) (in Russian).Google Scholar
Stekloff (V. A. Steklov), W., Problème de refroidissement d’une barre hétérogène. Ann. Fac. Sci. Toulouse Sér. 2 3 1901, 281–313.CrossRefGoogle Scholar
Stekloff (V. A. Steklov), W., Sur certaines égalités remarquables. C. R. Acad. Sci. Paris 135 1902, 783–786.Google Scholar
Stekloff (V. A. Steklov), W., Sur les problèmes fondamentaux de la physique mathematique (suite et fin). Ann. Sci. ENS Sér. 3 19 1902, 455–490.Google Scholar
Stekloff (V. A. Steklov), W., Sur la condition de fermeture des systèmes de fonctions orthogonales. C. R. Acad. Sci. Paris 151 1910, 1116–1119.Google Scholar
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, 1–87.Google Scholar
Strutt (Lord Rayleigh), J. W., The Theory of Sound, 2nd edn, Vo1. 1, Macmillan (London, 1894).Google Scholar
Szegő, G., Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3 1954, 343–356.Google Scholar
Talenti, G., Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 1976, 353–372.CrossRefGoogle Scholar
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, 19–46.Google Scholar
Valtorta, D., Sharp estimate on the first eigenvalue of the p-Laplacian. Nonlinear Anal. 75 2012, 4974–4994.CrossRefGoogle Scholar
Wang, X. J., Neumann problems of semilinear elliptic equations involving critical Sobolev exponents. J. Differential Equations 93(2) 1991, 283–310.CrossRefGoogle Scholar
Weinberger, H. F., An isoperimetric inequality for the n-dimensional free membrane problem. J. Ration. Mech. Anal. 5 1956, 633–636.Google Scholar
Weinstock, R., Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3 1954, 745–753.Google Scholar