Abstract

Let Ω\Omega be a bounded open convex set in Rn{\mathbb R}^n and let ϕ ⁣:Γ:=∂Ω→R\phi \colon \Gamma:=\partial\Omega\to {\mathbb R} be a function defined on its boundary. The lower bounded slope condition (on ϕ\phi) is a hypothesis recently introduced by F. Clarke [Ann. Scuola Norm. Sup. Pisa, in print], who has shown its relevance to regularity theory in the calculus of variations. It corresponds to a weaker version of the traditional bounded slope condition, which also appears in the theory of elliptic differential equations. In this paper, we study the regularity properties of these functions and give intrinsic characterizations of them. Semiconvexity turns out to be a central tool in the proofs

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References

  1. P. Bousquet: PhD Thesis, Université Claude Bernard Lyon 1 (2006).Google Scholar
  2. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  3. F. Clarke: Continuity of solutions to a basic problem in the calculus of variations, Ann. Sc. Norm. Sup. Pisa, Cl. Sci., V. Ser. 4 (2005) 511–530.https://doi.org/10.2422/2036-2145.2005.3.06Google Scholar
  4. F. Clarke: Multiple integrals of Lipschitz functions in the calculus of variations, Proc. Amer. Math. Soc. 64(2) (1977) 260–264.https://doi.org/10.1090/S0002-9939-1977-0451156-3Google Scholar
  5. F. Clarke, Y. S. Ledyaev, R. J. Stern, P. R. Wolenski: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).Google Scholar
  6. D. Gilbarg: Boundary value problems for nonlinear elliptic equations in n variables, in: Nonlinear Problems, Proc. Sympos. (Madison 1962), Univ. of Wisconsin Press, Madison (1963) 151–159.Google Scholar
  7. D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 Edition, Classics in Mathematics, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-61798-0Google Scholar
  8. E. Giusti: Direct Methods in the Calculus of Variations, World Scientific, River Edge (2003).https://doi.org/10.1142/5002Google Scholar
  9. A. Haar: Über das Plateausche Problem, Math. Ann 97 (1927) 124–258.https://doi.org/10.1007/BF01447864Google Scholar
  10. P. Hartman: On the bounded slope condition, Pacific J. Math. 18 (1966) 495–511.https://doi.org/10.2140/pjm.1966.18.495Google Scholar
  11. P. Hartman: On quasilinear elliptic functional-differential equations, Differ. Equations dynam. Systems, Proc. Int. Sympos. (Puerto Rico, 1965), Academic Press (1967) 393–407.Google Scholar
  12. P. Hartman: Convex sets and the bounded slope condition, Pacific J. Math. 25 (1968) 511–522.https://doi.org/10.2140/pjm.1968.25.511Google Scholar
  13. P. Hartman, L. Nirenberg: On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959) 901–920.https://doi.org/10.2307/2372995Google Scholar
  14. D. Hilbert: Über das Dirichletsche Prinzip, Jber. Dtsch. Math. Verein 8 (1900) 184–188.Google Scholar
  15. P. Marcellini: Regularity for elliptic equations with general growth conditions, J. Differ. Equations 105(2) (1993) 296–333.https://doi.org/10.1006/jdeq.1993.1091Google Scholar
  16. C. Mariconda, G. Treu: Existence and Lipschitz regularity for minima, Proc. Amer. Math. Soc. 130(2) (2002) 395–404.https://doi.org/10.1090/S0002-9939-01-06370-5Google Scholar
  17. C. Mariconda, G. Treu: Gradient maximum principle for minima, J. Optim. Theory Appl. 112(1) (2002) 167–186.https://doi.org/10.1023/A:1013052830852Google Scholar
  18. U. Massari, M. Miranda: Minimal Surfaces of Codimension One, North-Holland Mathematics Studies 91, North-Holland, Amsterdam (1984).Google Scholar
  19. M. Miranda: Un teorema di esistenza e unicità per il problema dell’area minima in n variabili, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19 (1965) 233–249.Google Scholar
  20. C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Grundlehren der mathematischen Wissenschaften 130, Springer, New York (1966).https://doi.org/10.1007/978-3-540-69952-1Google Scholar
  21. T. Rado: Geometrische Betrachtungen über zweidimensionale reguläre Variationsprobleme, Act. Lit. ac. Scient (Szeged) 2 (1924–1926) 228–253.Google Scholar
  22. T. Rado: On the Problem of Plateau, Erg. d. Math. u. ihrer Grenzgebiete 2, Springer, Berlin (1933).Google Scholar
  23. G. Stampacchia: On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963) 383–421.https://doi.org/10.1002/cpa.3160160403Google Scholar
  24. N. S. Trudinger: Lipschitz continuous solutions of elliptic equations of the form A(Du)D2 u = 0, Math. Z. 109 (1969) 211–216.https://doi.org/10.1007/BF01111406Google Scholar

Suggested citation

P. Bousquet. “On the Lower Bounded Slope Condition.” Journal of Convex Analysis 14 (2007), No. 1, 119–136. https://doi.org/10.68381/jca14010

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press.