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On a sub-supersolution method for the prescribed mean curvature problem

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Abstract

The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub-and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub-and supersolutions are established.

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References

  1. R. Acar and C. Vogel: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10 (1994), 1217–1229.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. Ambrosio, S. Mortola and V. Tortorelli: Functionals with linear growth defined on vector valued BV functions. J. Math. Pures Appl. 70 (1991), 269–323.

    MathSciNet  MATH  Google Scholar 

  3. G. Anzellotti and M. Giaquinta: BV functions and traces. Rend. Sem. Mat. Univ. Padova 60 (1978), 1–21.

    MathSciNet  MATH  Google Scholar 

  4. F. Atkinson, L. Peletier and J. Serrin: Ground states for the prescribed mean curvature equation: the supercritical case. Nonlinear Diffusion Equations and Their Equilibrium States, Math. Sci. Res. Inst. Publ., vol. 12, 1988, pp. 51–74.

  5. E. Bombieri and E. Giusti: Local estimates for the gradient of non-parametric surfaces of prescribed mean curvature. Comm. Pure Appl. Math. 26 (1973), 381–394.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Buttazzo: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Research Notes in Mathematics, vol. 207, Longman Scientific & Technical, Harlow, 1989.

    Google Scholar 

  7. S. Carl and V. K. Le: Sub-supersolution method for quasilinear parabolic variational inequalities. J. Math. Anal. Appl. 293 (2004), 269–284.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Carl, V. K. Le and D. Motreanu: Existence and comparison results for quasilinear evolution hemivariational inequalities. Electron. J. Differential Equations 57 (2004), 1–17.

    MathSciNet  Google Scholar 

  9. S. Carl, V. K. Le and D. Motreanu: The sub-supersolution method and extremal solutions for quasilinear hemivariational inequalities. Differential Integral Equations 17 (2004), 165–178.

    MathSciNet  MATH  Google Scholar 

  10. S. Carl, V. K. Le and D. Motreanu: Existence and comparison principles for general quasilinear variational-hemivariational inequalities. J. Math. Anal. Appl. 302 (2005), 65–83.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Carriero, A. Leaci and E. Pascali: On the semicontinuity and the relaxation for integrals with respect to the Lebesgue measure added to integrals with respect to a Radon measure. Ann. Mat. Pura Appl. 149 (1987), 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Carriero, Dal Maso, A. Leaci and E. Pascali: Relaxation of the nonparametric Plateau problem with an obstacle. J. Math. Pures Appl. 67 (1988), 359–396.

    MathSciNet  MATH  Google Scholar 

  13. C. V. Coffman and W. K. Ziemer: A prescribed mean curvature problem on domains without radial symmetry. SIAM J. Math. Anal. 22 (1991), 982–990.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Conti and F. Gazzola: Existence of ground states and free-boundary problems for the prescribed mean curvature equation. Adv. Differential Equations 7 (2002), 667–694.

    MathSciNet  MATH  Google Scholar 

  15. G. Dal-Maso: An introduction to Γ-convergence. Birkhäuser, 1993.

  16. I. Ekeland and R. Temam: Analyse convexe et problèmes variationnels. Dunod, 1974.

  17. L. C. Evans and R. F. Gariepy: Measure theory and fine properties of functions. CRC Press, Boca Raton, 1992.

    MATH  Google Scholar 

  18. R. Finn: Equilibrium capillary surfaces. Springer, New York, 1986.

    MATH  Google Scholar 

  19. F. Gastaldi and F. Tomarelli: Some remarks on nonlinear noncoercive variational inequalities. Boll. Un. Math. Ital. 7 (1987), 143–165.

    MathSciNet  Google Scholar 

  20. C. Gerhardt: Existence, regularity, and boundary behaviour of generalized surfaces of prescribed mean curvature. Math. Z. 139 (1974), 173–198.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Gerhardt: On the regularity of solutions to variational problems in BV (Ω). Math. Z. 149 (1976), 281–286.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Gerhardt: Boundary value problems for surfaces of prescribed mean curvature. J. Math. Pures Appl. 58 (1979), 75–109.

    MathSciNet  MATH  Google Scholar 

  23. D. Gilbarg and N. Trudinger: Elliptic partial differential equations of second order. Springer, Berlin, 1983.

    MATH  Google Scholar 

  24. E. Giusti: Minimal surfaces and functions of bounded variations. Birkhäuser, Basel, 1984.

    Google Scholar 

  25. C. Goffman and J. Serrin: Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964), 159–178.

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Habets and P. Omari: Positive solutions of an indefinite prescribed mean curvature problem on a general domain. Adv. Nonlinear Studies 4 (2004), 1–13.

    MathSciNet  MATH  Google Scholar 

  27. N. Ishimura: Nonlinear eigenvalue problem associated with the generalized capillarity equation. J. Fac. Sci. Univ. Tokyo 37 (1990), 457–466.

    MathSciNet  MATH  Google Scholar 

  28. T. Kusahara and H. Usami: A barrier method for quasilinear ordinary differential equations of the curvature type. Czech. Math. J. 50 (2000), 185–196.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. K. Le: Existence of positive solutions of variational inequalities by a subsolution-supersolution approach. J. Math. Anal. Appl. 252 (2000), 65–90.

    Article  MathSciNet  MATH  Google Scholar 

  30. V. K. Le: Subsolution-supersolution method in variational inequalities. Nonlinear Analysis 45 (2001), 775–800.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. K. Le: Some existence results on nontrivial solutions of the prescribed mean curvature equation. Adv. Nonlinear Studies 5 (2005), 133–161.

    MATH  Google Scholar 

  32. V. K. Le and K. Schmitt: Sub-supersolution theorems for quasilinear elliptic problems: A variational approach. Electron. J. Differential Equations (2004), 1–7.

  33. J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.

    MATH  Google Scholar 

  34. M. Marzocchi: Multiple solutions of quasilinear equations involving an area-type term. J. Math. Anal. Appl. 196 (1995), 1093–1104.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Mawhin and M. Willem: Critical point theory and Hamiltonian systems. Springer Verlag, New York, 1989.

    MATH  Google Scholar 

  36. M. Miranda: Dirichlet problem with L 1 data for the non-homogeneous minimal surface equation. Indiana Univ. Math. J. 24 (1974), 227–241.

    Article  MathSciNet  MATH  Google Scholar 

  37. W. M. Ni and J. Serrin: Existence and non-existence theorems for quasilinear partial differential equations the anomalous case. Accad. Naz. Lincei, Atti dei Convegni 77 (1985), 231–257.

    Google Scholar 

  38. W. M. Ni and J. Serrin: Non-existence theorems for quasilinear partial differential equations. Rend. Circ. Math. Palermo 2 (1985), 171–185.

    MathSciNet  Google Scholar 

  39. E. S. Noussair, C. A. Swanson and Y. Jianfu: A barrier method for mean curvature problems. Nonlinear Anal. 21 (1993), 631–641.

    Article  MathSciNet  MATH  Google Scholar 

  40. L. Peletier and J. Serrin: Ground states for the prescribed mean curvature equation. Proc. Amer. Math. Soc. 100 (1987), 694–700.

    Article  MathSciNet  MATH  Google Scholar 

  41. W. Ziemer: Weakly differentiable functions. Springer, New York, 1989.

    MATH  Google Scholar 

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  1. Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO, 65409, USA

    Vy khoi Le

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Le, V.k. On a sub-supersolution method for the prescribed mean curvature problem. Czech Math J 58, 541–560 (2008). https://doi.org/10.1007/s10587-008-0034-7

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