The Neumann eigenvalue problem for the p(x) -Laplacian as p → ∞
Keywords:
∞-Laplacian, eigenvalue problems, Luxemburg norm, p(x)-Laplacian, variable exponent Lebesgue and Sobolev spaces, viscosity solutionsAbstract
This paper is dedicated to the study of the behaviour of the second eigenvalues and the corresponding eigenfunctions for the p(x)-Laplacian subject to the Neumann boundary conditions in an open, bounded domain Ω ⊂ RN with smooth boundary. As p → ∞ one can obtain uniform bounds for the sequence of second eigenvalues and the positive second eigenfunctions. In the latter case, the uniform limit is a viscosity solution to a problem involving the ∞-Laplacian subject to appropriate boundary conditions.
References
F. Abdullayev, M. Bocea, The Robin eigenvalue problem for the p(x)-Laplacian as p → ∞. Nonlinear Anal. 91
(2013), 32-45
E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech.
Anal. 156 (2001), 121-140.
G. Barles, Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differential Equations 106 (1993), 90-106.
F. Browder, Existence theorems for nonlinear partial differential equations. In: Global Analysis, 1-60 (Proceedings of the Symposium Pure Mathematics, Vol. 16, Berkeley, California, 1968), Amer. Math. Soc., Providence, RI, 1970.
M. G. Crandall, H. Ishii, and P.L. Lions, User’s guide to viscosity solutions of second-order partial differential equations. Bull. Am. Math. Soc. 27 (1992), 1-67.
Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math
(2006), 1383-1406.
S.-G. Deng, Q. Wang, S. Cheng, On the p(x)-Laplacian Robin eigenvalue problem. Appl. Math. Comput. 217 No.
(2011), 5643-5649.
S.-G. Deng, Eigenvalues of the p(x)-Laplacian Steklov problem, J. Math. Anal. Appl. 339 (2008), 925-937. [9] E. DiBenedetto, Degenerate Parabolic Equations. Springer-Verlag, New York, 1993.
L. Diening, P. Harjulehto, P. H¨asto¨, and M. Ruˇziˇcka, Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011.
D. E. Edmunds, J. Lang, and A. Nekvinda, On Lp(x) norms. Proc. Roy. Soc. London Ser. A 455 (1999), 219-225. [12] D. E. Edmunds and J. R´akosn´ık, Density of smooth functions in Wk,p(x)(Ω). Proc. R. Soc. Lond. Ser. A 437 (1992),
-236.
D. E. Edmunds and J. R´akosn´ık, Sobolev embeddings with variable exponent. Studia Math. 143 (2000), 267-293. [14] X.-L. Fan, Y.Z. Zhao, Q.-H. Zhang, A strong maximum principle for p(x)-Laplace equations. Chinese J. Contemp.
Math. 24 (2003), 277-282.
X.-L. Fan, Q.-H. Zhang, D. Zhao, Eigenvalues of the p(x)-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302
(2005), 306-317.
X.-L. Fan, Remarks on eigenvalue problems involving the p(x)-Laplacian. J. Math. Anal. Appl. 352 No. 1 (2009), 85-98.
X.-L. Fan, Eigenvalues of the p(x)-Laplacian Neumann problems. Nonlinear Analysis 67 (2007), 2982-2992.
G. Franzina and P. Lindqvist, An eigenvalue problem with variable exponents. Nonlinear Anal. 85 (2013), 1-16. [19] O. Kov´aˇcik and J. R´akosn´ık, On spaces Lp(x) and W 1,p(x). Czechoslovak Math. J. 41 (1991), 592-618.
P. Lindqvist and T. Lukkari, A curious equation involving the ∞-Laplacian. Adv. Calc. Var. 3 No. 4 (2010), 409-421.
J. J. Manfredi, J. D. Rossi, and J. M. Urbano, p(x)-harmonic functions with unbounded exponent in a subdomain.
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 26 No. 6 (2009), 2581-2595.
J. J. Manfredi, J. D. Rossi, and J. M. Urbano, Limits as p(x) → ∞ of p(x)-harmonic functions. Nonlinear Anal. 72
(2010), 309-315.
M. Miha˘ilescu, Eigenvalue Problems for Some Elliptic Partial Differential Operators. Ph.D. Thesis, Central European University, Budapest, 2010. Available online at:
http://web.ceu.hu/math/People/Alumni and Friends/Alumni/Mihai MIHAILESCU Thesis.pdf
J. Musielak, Orlicz spaces and modular spaces. Lecture notes in mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
M. P´erez-Llanos and J.D. Rossi, The behaviour of the p(x)-Laplacian eigenvalue problem as p(x) → ∞. J. Math.
Anal. Appl. 363 (2010), 502-511.
M. P´erez-Llanos and J. D. Rossi, Limits as p(x) → ∞ of p(x)-harmonic functions with non-homogeneous Neumann boundary conditions. In: Nonlinear elliptic partial differential equations, 187-201, Contemp. Math., Vol. 540, Amer. Math. Soc., Providence, RI, 2011.
M. Ruzicka, Electrorheological fluids: modeling and mathematical theory. Springer-Verlag, Berlin (2002).
L. Yu, F. Li, The stability of eigenvalues for the p(x)-Laplacian equations involving Robin boundary conditions, J. Dyn. Con. Sys. 24 (2018), 223-236.
E. Zeidler, Variational Methods and Optimization. Nonlinear Functional Analysis and its Applications, vol. 3, Springer-Verlag, Berlin, 1990.
V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33-36.
