Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a family of inhomogeneous torsional creep problems

Authors: Marian Bocea and Mihai Mihăilescu
Journal: Proc. Amer. Math. Soc. 145 (2017), 4397-4409
MSC (2010): Primary 35D30, 35D40, 46E30, 46E35, 49J40, 49J45, 49S99
Published electronically: May 4, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The asymptotic behavior of solutions to a family of Dirichlet boundary value problems involving inhomogeneous PDEs in divergence form is studied in an Orlicz-Sobolev setting. Solutions are shown to converge uniformly to the distance function to the boundary of the domain. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold under much more general constitutive assumptions on the stress.

References [Enhancements On Off] (What's this?)

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] T. Bhattacharya, E. DiBenedetto, and J. Manfredi, Limits as $ p\to \infty $ of $ \Delta _pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1989), 15-68 (1991). Some topics in nonlinear PDEs (Turin, 1989). MR 1155453
  • [3] Marian Bocea and Mihai Mihăilescu, $ \Gamma $-convergence of inhomogeneous functionals in Orlicz-Sobolev spaces, Proc. Edinb. Math. Soc. (2) 58 (2015), no. 2, 287-303. MR 3341440,
  • [4] Marian Bocea, Mihai Mihăilescu, and Denisa Stancu-Dumitru, The limiting behavior of solutions to inhomogeneous eigenvalue problems in Orlicz-Sobolev spaces, Adv. Nonlinear Stud. 14 (2014), no. 4, 977-990. MR 3269381,
  • [5] Philippe Clément, Ben de Pagter, Guido Sweers, and François de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math. 1 (2004), no. 3, 241-267. MR 2094464,
  • [6] Nobuyoshi Fukagai, Masayuki Ito, and Kimiaki Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $ \mathbf {R}^N$, Funkcial. Ekvac. 49 (2006), no. 2, 235-267. MR 2271234,
  • [7] Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51-74. MR 1218686,
  • [8] Jürgen Jost and Xianqing Li-Jost, Calculus of variations, Cambridge Studies in Advanced Mathematics, vol. 64, Cambridge University Press, Cambridge, 1998. MR 1674720
  • [9] Petri Juutinen, Peter Lindqvist, and Juan J. Manfredi, The $ \infty $-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 89-105. MR 1716563,
  • [10] L. M. Kachanov, The theory of creep, Nat. Lending Lib. for Science and Technology, Boston Spa, Yorkshire, England, 1967.
  • [11] L. M. Kachanov, Foundations of the theory of plasticity, Translated from the Russian second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971. North-Holland Series in Applied Mathematics and Mechanics, Vol. 12. MR 0483881
  • [12] Ryuji Kajikiya, A priori estimate for the first eigenvalue of the $ p$-Laplacian, Differential Integral Equations 28 (2015), no. 9-10, 1011-1028. MR 3360728
  • [13] Bernhard Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. 410 (1990), 1-22. MR 1068797,
  • [14] Gary M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311-361. MR 1104103,
  • [15] L. E. Payne and G. A. Philippin, Some applications of the maximum principle in the problem of torsional creep, SIAM J. Appl. Math. 33 (1977), no. 3, 446-455. MR 0455738
  • [16] Sandra Martínez and Noemi Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 (2008), no. 6, 1914-1971. MR 2431665,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35D30, 35D40, 46E30, 46E35, 49J40, 49J45, 49S99

Retrieve articles in all journals with MSC (2010): 35D30, 35D40, 46E30, 46E35, 49J40, 49J45, 49S99

Additional Information

Marian Bocea
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660

Mihai Mihăilescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania — and — “Simion Stoilow” Institute of Mathematics of the Romanian Academy, 010702 Bucharest, Romania

Keywords: Inhomogeneous equations, Orlicz-Sobolev spaces, torsional creep, viscosity solutions
Received by editor(s): April 28, 2016
Received by editor(s) in revised form: November 9, 2016
Published electronically: May 4, 2017
Additional Notes: The research of the first author was partially supported by the U.S. National Science Foundation under Grant No. DMS-1515871. The second author was partially supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE-2016-0035.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society