On a family of inhomogeneous torsional creep problems

Bocea, Marian; Mihăilescu, Mihai

doi:10.1090/proc/13583

On a family of inhomogeneous torsional creep problems
HTML articles powered by AMS MathViewer

by Marian Bocea and Mihai Mihăilescu

Proc. Amer. Math. Soc. 145 (2017), 4397-4409

DOI: https://doi.org/10.1090/proc/13583

Published electronically: May 4, 2017

PDF | Request permission

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

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
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
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, DOI 10.1017/S0013091514000170
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, DOI 10.1515/ans-2014-0409
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, DOI 10.1007/s00009-004-0014-6
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, DOI 10.1619/fesi.49.235
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, DOI 10.1007/BF00386368
Jürgen Jost and Xianqing Li-Jost, Calculus of variations, Cambridge Studies in Advanced Mathematics, vol. 64, Cambridge University Press, Cambridge, 1998. MR 1674720
Petri Juutinen, Peter Lindqvist, and Juan J. Manfredi, The $\infty$-eigenvalue problem, Arch. Ration. Mech. Anal. 148 (1999), no. 2, 89–105. MR 1716563, DOI 10.1007/s002050050157
L. M. Kachanov, The theory of creep, Nat. Lending Lib. for Science and Technology, Boston Spa, Yorkshire, England, 1967.
L. M. Kachanov, Foundations of the theory of plasticity, Translated from the Russian second revised edition, North-Holland Series in Applied Mathematics and Mechanics, Vol. 12, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1971. MR 0483881
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
Bernhard Kawohl, On a family of torsional creep problems, J. Reine Angew. Math. 410 (1990), 1–22. MR 1068797, DOI 10.1515/crll.1990.410.1
Gary M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural′tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311–361. MR 1104103, DOI 10.1080/03605309108820761
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 455738, DOI 10.1137/0133028
Sandra Martínez and Noemi Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 (2008), no. 6, 1914–1971. MR 2431665, DOI 10.1016/j.aim.2008.03.028