Skip to main content
SpringerLink
Log in
Book cover

The Maz’ya Anniversary Collection pp 249–306Cite as

Birkhäuser

Sharp constants and maximum principles for elliptic and parabolic systems with continuous boundary data

  • Conference paper

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 109))

Abstract

This review is concerned with results obtained in a series of joint papers by V. Maz’ya and the author, and it covers a number of topics. First, explicit formulae for essential norms of the elastic and hydrodynamic double layer potentials are discussed for boundaries, having vertices and edges, and these norms are considered in the space of continuous vector-valued functions. Secondly, the explicit expressions for the best constants in estimates of solutions of some systems and equations of mathematical physics are surveyed. Third, criteria for the validity of the maximum modulus principle are stated for solutions of elliptic and parabolic systems. The fourth topic deals with necessary and sufficient conditions providing the maximum norm principle for parabolic systems. Here the norm is defined as Minkowski’ s functional of a convex body.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agmon, S., Douglis, A. and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–737.

    Article  MathSciNet  MATH  Google Scholar 

  2. Agmon, S., Maximum theorems for solutions of higher order elliptic equations, Bull. Amer. Math. Soc. 66 (1960), 77–80.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Amann, H., Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl. 65 (1978), 432–467.

    Article  MathSciNet  MATH  Google Scholar 

  5. Angell, T.S., Kleinman, R.E. and KrÁl, J., Layer potentials on boundaries with corners and edges, Časopis pro pěstovánÍ matematiky 113(4) (1988), 387–402.

    Google Scholar 

  6. Bebernes, J. and Smitt, K., Invariant sets and the Hukuhara-Kneser property for systems of parabolic partial differential equations, Rocky Mountain J. Math. 7 (1977), 557–567.

    Article  MathSciNet  MATH  Google Scholar 

  7. BlTsadze, A.V., On elliptic systems of second order partial differential equations, Dokl. Akad. Nauk SSSR 112 (1957), 983–986 (Russian).

    MathSciNet  Google Scholar 

  8. Burago, Ju.D. and Maz’ya, V.G., Multidimensional potential theory and the solution of the boundary value problems for regions with irregular boundaries, Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklov (LOMI) 3, Part 1 (1967), 1–86; English translation: Sem. V.A. Steklov Math. Inst., Leningrad 3 (1967), 3-39.

    Google Scholar 

  9. Carleman, T., Über das Neumann-Poincaresche Problem für ein Gebiet mit Ecken, Inaugural Dissertation, Tr. Almqvist and Wiksell, Uppsala, 1916.

    Google Scholar 

  10. Chueh, K.N., Conley C.C. and Smoller, J.A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), 373–391.

    Article  MathSciNet  MATH  Google Scholar 

  11. Cosner, C. and Schaeffer, P.W., On the development of functionals which satisfy a maximum principle, Appl. Anal. 26 (1987), 45–60.

    Article  MathSciNet  MATH  Google Scholar 

  12. Costabel, M., Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988), 613–626.

    Article  MathSciNet  MATH  Google Scholar 

  13. Eidel’man, S.D., Parabolic Systems, North-Holland, Amsterdam, 1969.

    Google Scholar 

  14. Elschner, J., The double layer potential operator over polyhedral domains. I: Solvability in weighted Sobolev spaces, Appl. Analysis 45 (1992), 117–134.

    Article  MathSciNet  MATH  Google Scholar 

  15. Fabes, E., Layer potential methods for boundary value problems on Lipschitz domains, in: Potential theory. Survey and problems, Proceedings, Prague 1987 J. Král, J. Lukeš, I. Netuka and J. Veselý (eds.) Lecture Notes in Mathematics, 1344, Springer, Berlin-Heidelberg-New York, 1988.

    Google Scholar 

  16. Federer, H., Geometric Measure Theory, Springer, Berlin-Heidelberg-New York, 1969.

    MATH  Google Scholar 

  17. Germain, P., Course de Mécanique des Milieux Continus, t. I. Théorie Générale, Masson et Cje, Ȥiteurs, Paris, 1973.

    Google Scholar 

  18. Gohberg, I. and Marcus, A., Some remarks on topologically equivalent norms, Izvestia Mold. Fil. Akad. Nauk SSSR 10(76) (1960), 91–94.

    Google Scholar 

  19. Grachev, N.V., Representation and estimates for inverse operators of the potential theory integral equations in a polyhedron, in: Potential Theory, Proc Int. Conf. on Potential Theory, Nagoya, 1990, (Masanori Kishi, ed.), 201-206.

    Google Scholar 

  20. Grachev, N.V. and Maz’ya, V.G., On the Fredholm radius for operators of the double layer potential type on piecewise smooth boundaries, Vestnik Leningr. Univ. 19(4) (1986), 60–64 (in Russian).

    MathSciNet  Google Scholar 

  21. Grachev, N.V. and Maz’ya, V.G., Solvability of a boundary integral equation on a polyhedron, Report LiTH-MAT-R-91-50, Linköping Univ., Sweden.

    Google Scholar 

  22. Hile, G. and Protter, M.H., Maximum principles for a class of first-order elliptical systems, J. Diff. Equations 24 (1977), 136–151.

    Article  MathSciNet  MATH  Google Scholar 

  23. KrÁl, J., The Fredholm method in potential theory, Trans. Amer. Math. Soc. 125(3) (1966), 511–547.

    Article  MathSciNet  MATH  Google Scholar 

  24. KrÁl, J. and Wendland, W.L., Some examples concerning applicability of the Fredholm-Radon method in potential theory, Aplikace Matematiky 31 (1986), 293–298.

    MathSciNet  MATH  Google Scholar 

  25. KrÁl, J. and Wendland, W.L., On the applicability of the Fredholm-Radon method in potential theory and the panel method, in: Panel methods in fluid mechanics with emphasis in aero-dynamics, (J. Ballmann, R. Appier, W. Hackbush, eds.), Notes on numerical Fluid Mechanics 21, Vieweg, Braunschweig, 1988, 120–136.

    Google Scholar 

  26. Kresin, G.I. and Maz’ya, V.G., On the maximum of displacements in the visco-elastic half-space (three parametrical model), Vestnik Leningr. Univ. 22 (1985), 47–51 (Russian).

    MathSciNet  Google Scholar 

  27. Kresin, G.I. and Maz’ya, V.G., On the exact constant in the inequality of Miranda-Agmon type for solutions of elliptic equations, Izvestia Vys. Uch. Zaved., Matem. Ser. 5 (1988), 41–50 (Russian).

    MathSciNet  Google Scholar 

  28. Kresin, G.I. and Maz’ya, V.G., Criteria for validity of the maximum modulus principle for solutions of linear strongly elliptic systems, Potential Analysis 2 (1993), 73–99.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kresin, G.I., On the exact constant in the inequality for a norm of solutions of the Lamé system in a half-space, Functional Differential Equations. Israel Seminar. The College of Judea and Samaria 1 (1993), 132–135.

    MathSciNet  MATH  Google Scholar 

  30. Kresin, G.I. and Maz’ya, V.G., Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems, Ark. Math. 32 (1994), 121–155.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kresin, G.I. and Maz’ya, V.G., The norm and the essential norm of the double layer elastic and hydrodynamics potentials in the space of continuous functions, Math. Meth. in Appl. Sc. 18 (1995), 1095–1131.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kresin, G.I. and Maz’ya, V.G., Criteria for validity of the maximum norm principle for parabolic systems, Potential Analysis, to appear.

    Google Scholar 

  33. Kresin, G.I. and Maz’ya, V.G., On the maximum principle with respect to smooth norms for linear strongly coupled parabolic systems, Functional Differential Equations, to appear.

    Google Scholar 

  34. Kresin, G.I. and Maz’ya, V.G., On the maximum modulus principle for linear parabolic systems with zero boundary data, Functional Differential Equations 5 (1998), 165–181.

    MathSciNet  MATH  Google Scholar 

  35. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O. and Burchuladze T.V., Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, 2nd ed., Nauka, Moscow, 1976; English transl.: North-Holland, 1979.

    Google Scholar 

  36. Ladyzhenskaya, O.A., Mathematical questions in the hydrodynamics of a viscous incompressible fluid, 2nd ed., Nauka, Moscow, 1970; English transl, of 1st ed., The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1963.

    Google Scholar 

  37. Maz’ya, V.G., Boundary integral equations of elasticity in domains with piecewise smooth boundaries, in: EQUADIFF 6 Proa, Brno Univ., 1985, 235–242.

    Google Scholar 

  38. Maz’ya, V.G., Boundary Integral Equations, in: Encyclopedia of Math. Sc, 13, Springer, Berlin-Heidelberg-New York, 1991.

    Google Scholar 

  39. Maz’ya, V.G. and Kresin, G.I., On the maximum principle for strongly elliptic and parabolic second order systems with constant coefficients, Mat. Sb. 125 (1984), 458–480 (Russian). English transl.: Math. USSR-Sb. 53 (1986), 457-479.

    MathSciNet  Google Scholar 

  40. Mcmullen, P. and Sheppard, G.C., Convex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lect. Notes Series, 3, Cambridge Univ. Press, 1971.

    Google Scholar 

  41. Minkowski, H., Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs. Posthum. Gesammelte Abhandlungen von Herman Minkowski (ed. D. Hilbert), vol. 2, Leipzig: Teubner, 1911.

    Google Scholar 

  42. Miranda, C., Formule di maggiorazione e teorema di esistenza per 1e funzioni biarmoniche di due variabili, Giorn. Mat. Bataglini 78 (1948-1949), 97–118.

    MathSciNet  MATH  Google Scholar 

  43. Miranda, C., Teorema del massimo modulo e teorema di esistenza e di unicità per il problema di Dirichlet relativo alle equazioni ellitiche in due variabili, Ann. Mat. Pura Appl, ser. 4 46 (1958), 265–311.

    Article  MathSciNet  MATH  Google Scholar 

  44. Miranda, C., Sul teorema del massimo modulo per una classe di sistemi ellitici di equazioni del secondo ordine e per 1e equazioni a coefficienti complessi, Istit. Lombardo Accad. Sci. Lett. Rend. A 104 (1970), 736–745.

    MathSciNet  MATH  Google Scholar 

  45. Parton, V.Z. and Perlin P.I., Methods in the mathematical theory of elasticity, Nauka, Moscow, 1981 (Russian).

    Google Scholar 

  46. Pini, B., Sui sistemi equazioni lineari a derivate parziali del secondo ordine dei tipi ellittico e parabolico, Rend, del Sem. Mat. della Univ. di Padova 22 (1953), 265–280.

    MathSciNet  MATH  Google Scholar 

  47. Polya, G., Liegt die Stelle der größten Beanspruchungen an der Oberfläche?, Z. Angew. Math. Mech., 10 (1930), 353–360.

    Article  MATH  Google Scholar 

  48. Protter, M.H. and Weinberger, H.F., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967; Springer, Berlin-Heidelberg-New York, 1985.

    Google Scholar 

  49. Protter, M.H., Maximum principles, Pittman Research Notes, Math. ser. 175 (1988), 1–14.

    MathSciNet  Google Scholar 

  50. Radon, J., Über lineare Functionaltransformationen und Functionalgleichungen, Sitzungsber. Akad. Wiss. Wien, Abt. 2a, 128, H. 7, (1919), 1083–1121.

    Google Scholar 

  51. Radon, J., Über die Randwertaufgaben beim logarithmischen Potential, Sitzungsber. Akad. Wiss. Wien, Abt. 2a, 128, H. 7, (1919), 1123–1167.

    Google Scholar 

  52. Rathsfeld, A., The invertibility of the double layer potential operator in the space of continuous functions defined on a polyhedron: The panel method, Appl. Analysis 45 (1992), 135–177.

    Article  MathSciNet  MATH  Google Scholar 

  53. Redheffer, R. and Walter, W., Invariant sets for systems of partial differential equations. I. Parabolic equations, Arch. Rational Mech. Anal. 67 (1978), 41–52.

    Article  MathSciNet  Google Scholar 

  54. Rempel, S. and Schmidt, G., Eigenvalues for spherical domains with corners via boundary integral equations, Int. Equat. Oper. Theory 14 (1991), 229–250.

    Article  MathSciNet  MATH  Google Scholar 

  55. Shapiro, Z.Ya., The first boundary value problem for an elliptic system of differential equations, Mat. Sb. 28(70) (1951), 55–78 (Russian).

    Google Scholar 

  56. Shelepov, V. Yu., Investigations of matrix integral equations in the space of continuous functions by Ya. B. Lopatinskii’s method, in: General theory of boundary value problems. Collect. Sci. Works, Naukova dumka, Kiev, 1983, 220–226 (Russian).

    Google Scholar 

  57. Solonnikov, V.A., On general boundary value problems for systems elliptic in the Douglis-Nirenberg sense, Izv. Akad. Nauk SSSR, ser. Mat. 28 (1964), 665–706; English transl, in: Amer. Math. Soc. Transl.(2) 56 (1966).

    MathSciNet  MATH  Google Scholar 

  58. Stoer, J. and WlTzgall, C., Convexity and Optimization in Finite Dimensions I, Springer, Berlin-Heidelberg-New York, 1970.

    Book  Google Scholar 

  59. Stys, T., On the unique solvability of the first Fourier problem for a parabolic system of second order linear differential equations, Comment. Math. Prace Mat. 9 (1965), 282–283 (Russian).

    MathSciNet  Google Scholar 

  60. Szeptycki, P., Existence theorem for the first boundary value problem for a quasi-linear elliptic system, Bulletin Acad. Polon. des Sciences 7 (1959), 419–424.

    MathSciNet  MATH  Google Scholar 

  61. Verchota, G.C., Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 419–424.

    Article  MathSciNet  Google Scholar 

  62. Walter, W., Differential and Integral Inequalities, Springer, Berlin-Heidelberg-New York, 1970.

    MATH  Google Scholar 

  63. Wasowski, J., Maximum principle for a certain strongly elliptic system of linear equations of second order, Bull. Acad. Polon. Sci., 18 (1970), 741–745.

    MathSciNet  MATH  Google Scholar 

  64. Weinberger, H.F., Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. 8 (1975), 295–310.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute, The Colledge of Judea and Samaria, 44837, Ariel, Israel

    Gershon Kresin

Authors
  1. Gershon Kresin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Rostock, D-18051, Rostock, Germany

    Jürgen Rossmann , Peter Takáč  & Günther Wildenhain ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Basel AG

About this paper

Cite this paper

Kresin, G. (1999). Sharp constants and maximum principles for elliptic and parabolic systems with continuous boundary data. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8675-8_19

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9726-6

  • Online ISBN: 978-3-0348-8675-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation