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On the stability and transition of the Cahn-Hilliard/Allen-Cahn system
The asymptotic behaviour of the $ p(x) $-Laplacian Steklov eigenvalue problem
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China |
In this paper we study the asymptotic behaviour of the first eigenvalues $ \lambda^{1}_{p_{n}(\cdot)} $ and the corresponding eigenfunctions $ u_{n} $ of (1) as $ p_{n}(x)\rightarrow \infty $. Under adequate hypotheses on the sequence $ p_{n} $, we prove that $ \lambda^{1}_{p_{n}(\cdot)} $ converges to 1 and the positive first eigenfunctions $ u_{n} $, normalized by $ |u_{n}|_{L^{p_{n}(x)}(\partial \Omega)} = 1 $, converge, up to subsequences, to $ u_{\infty} $ uniformly in $ C^{\alpha}(\overline{\Omega}) $, for some $ 0<\alpha<1 $, where $ u_{\infty} $ is a nontrivial viscosity solution of a problem involving the $ \infty $-Laplacian subject to appropriate boundary conditions.
References:
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F. Abdullayev and M. Bocea,
The Robin eigenvalue problem for the $p(x)$-Laplacian as $p\rightarrow\infty$, Nonlinear Anal., 91 (2013), 32-45.
doi: 10.1016/j.na.2013.06.005. |
| [2] |
E. Acerbi and G. Mingione,
Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.
doi: 10.1007/s002050100117. |
| [3] |
G. Barles,
Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations, 106 (1993), 90-106.
doi: 10.1006/jdeq.1993.1100. |
| [4] |
M. Bocea and M. Mihǎilescu,
The principal frequency of $\triangle_{\infty}$ as a limit of Rayleigh quotients involving Luxemburg norms, Bull. Sci.math., 138 (2014), 236-252.
doi: 10.1016/j.bulsci.2013.06.001. |
| [5] |
M. Bocea and M. Mihǎilescu,
$\Gamma$-convergence of power-law functionals with variable exponents, Nonlinear Anal., 73 (2010), 110-121.
doi: 10.1016/j.na.2010.03.004. |
| [6] |
Y. M. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
| [7] |
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.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [8] |
S. G. Deng,
Eigenvalues of the $p(x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.
doi: 10.1016/j.jmaa.2007.07.028. |
| [9] |
L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
| [10] |
D. E. Edmunds and J. Rákosník,
Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.
doi: 10.4064/sm-143-3-267-293. |
| [11] |
X. L. Fan and X. Han,
Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^{N}$, Nonlinear Anal., 59 (2004), 173-188.
doi: 10.1016/j.na.2004.07.009. |
| [12] |
X. L. Fan, J. S. Shen and D. Zhao,
Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
doi: 10.1006/jmaa.2001.7618. |
| [13] |
X. L. Fan and D. Zhao,
On the Spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
| [14] |
X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)$-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277–282. (Translation of Chinese Ann. Math. Ser. A, 24 (2003) 495–500.) |
| [15] |
G. Franzina and P. Lindqvist,
An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16.
doi: 10.1016/j.na.2013.02.011. |
| [16] |
N. Fukagai, M. Ito and K. Narukawa,
Limit as $p\rightarrow\infty$ of $p$-Laplace eigenvalue problems and $L^{\infty}$ inequality of the Poincaré type, Differ. Integral Equations, 12 (1999), 183-206.
|
| [17] |
P. Harjulehto, P. Hästö, Ú. Lê and M. Nuortio,
Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
| [18] |
P. Juutinen, P. Lindqvist and J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
| [19] |
O. Kováčik and J. Rákosník,
On spaces $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$, Czechoslovak Math. J, 41 (1991), 592-618.
|
| [20] |
A. Lê,
On the first engenvalue of the Steklov eigenvalue problem for the infinity Laplacian, Electron. J. Differential Equations, 2006 (2006), 1-9.
|
| [21] |
P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, 102. University of Jyväskylä, Jyväskylä, 2006. |
| [22] |
P. Lindqvist and T. Lukkari,
A curious equation involving the $\infty$-Laplacian, Adv. Calc. Var., 3 (2010), 409-421.
|
| [23] |
J. J. Manfredi, J. D. Rossi and J. M. Urbano,
Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315.
doi: 10.1016/j.na.2009.06.054. |
| [24] |
J. J. Manfredi, J. D. Rossi and J. M. Urbano,
$p(x)$-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595.
doi: 10.1016/j.anihpc.2009.09.008. |
| [25] |
J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
| [26] |
M. Pérez-Llanos and J. D. Rossi, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Nonlinear Elliptic Partial Differential Equations, 187–201, Contemp. Math., 540, Amer. Math. Soc., Providence, RI, 2011.
doi: 10.1090/conm/540/10665. |
| [27] |
M. Pérez-Llanos and J. D. Rossi,
The behaviour of the $p(x)$-Laplacian eigenvalue problem as $p(x)\rightarrow \infty$, J. Math. Anal. Appl., 363 (2010), 502-511.
doi: 10.1016/j.jmaa.2009.09.044. |
| [28] |
M. Råžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
| [29] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 29 (1987), 33-66.
|
| [30] |
V. V. Zhikov,
On Lavrentiev$'$s phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269.
|
show all references
References:
| [1] |
F. Abdullayev and M. Bocea,
The Robin eigenvalue problem for the $p(x)$-Laplacian as $p\rightarrow\infty$, Nonlinear Anal., 91 (2013), 32-45.
doi: 10.1016/j.na.2013.06.005. |
| [2] |
E. Acerbi and G. Mingione,
Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal., 156 (2001), 121-140.
doi: 10.1007/s002050100117. |
| [3] |
G. Barles,
Fully non-linear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations, 106 (1993), 90-106.
doi: 10.1006/jdeq.1993.1100. |
| [4] |
M. Bocea and M. Mihǎilescu,
The principal frequency of $\triangle_{\infty}$ as a limit of Rayleigh quotients involving Luxemburg norms, Bull. Sci.math., 138 (2014), 236-252.
doi: 10.1016/j.bulsci.2013.06.001. |
| [5] |
M. Bocea and M. Mihǎilescu,
$\Gamma$-convergence of power-law functionals with variable exponents, Nonlinear Anal., 73 (2010), 110-121.
doi: 10.1016/j.na.2010.03.004. |
| [6] |
Y. M. Chen, S. Levine and M. Rao,
Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
| [7] |
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.
doi: 10.1090/S0273-0979-1992-00266-5. |
| [8] |
S. G. Deng,
Eigenvalues of the $p(x)$-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.
doi: 10.1016/j.jmaa.2007.07.028. |
| [9] |
L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
| [10] |
D. E. Edmunds and J. Rákosník,
Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.
doi: 10.4064/sm-143-3-267-293. |
| [11] |
X. L. Fan and X. Han,
Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^{N}$, Nonlinear Anal., 59 (2004), 173-188.
doi: 10.1016/j.na.2004.07.009. |
| [12] |
X. L. Fan, J. S. Shen and D. Zhao,
Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
doi: 10.1006/jmaa.2001.7618. |
| [13] |
X. L. Fan and D. Zhao,
On the Spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
| [14] |
X. L. Fan, Y. Z. Zhao and Q. H. Zhang, A strong maximum principle for $p(x)$-Laplace equations, Chinese J. Contemp. Math., 24 (2003), 277–282. (Translation of Chinese Ann. Math. Ser. A, 24 (2003) 495–500.) |
| [15] |
G. Franzina and P. Lindqvist,
An eigenvalue problem with variable exponents, Nonlinear Anal., 85 (2013), 1-16.
doi: 10.1016/j.na.2013.02.011. |
| [16] |
N. Fukagai, M. Ito and K. Narukawa,
Limit as $p\rightarrow\infty$ of $p$-Laplace eigenvalue problems and $L^{\infty}$ inequality of the Poincaré type, Differ. Integral Equations, 12 (1999), 183-206.
|
| [17] |
P. Harjulehto, P. Hästö, Ú. Lê and M. Nuortio,
Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
| [18] |
P. Juutinen, P. Lindqvist and J. Manfredi,
The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105.
doi: 10.1007/s002050050157. |
| [19] |
O. Kováčik and J. Rákosník,
On spaces $L^{p(x)}(\Omega)$ and $W^{k, p(x)}(\Omega)$, Czechoslovak Math. J, 41 (1991), 592-618.
|
| [20] |
A. Lê,
On the first engenvalue of the Steklov eigenvalue problem for the infinity Laplacian, Electron. J. Differential Equations, 2006 (2006), 1-9.
|
| [21] |
P. Lindqvist, Notes on the $p$-Laplace Equation, Report. University of Jyväskylä Department of Mathematics and Statistics, 102. University of Jyväskylä, Jyväskylä, 2006. |
| [22] |
P. Lindqvist and T. Lukkari,
A curious equation involving the $\infty$-Laplacian, Adv. Calc. Var., 3 (2010), 409-421.
|
| [23] |
J. J. Manfredi, J. D. Rossi and J. M. Urbano,
Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions, Nonlinear Anal., 72 (2010), 309-315.
doi: 10.1016/j.na.2009.06.054. |
| [24] |
J. J. Manfredi, J. D. Rossi and J. M. Urbano,
$p(x)$-harmonic functions with unbounded exponent in a subdomain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2581-2595.
doi: 10.1016/j.anihpc.2009.09.008. |
| [25] |
J. Musielak, Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
doi: 10.1007/BFb0072210. |
| [26] |
M. Pérez-Llanos and J. D. Rossi, Limits as $p(x)\rightarrow\infty$ of $p(x)$-harmonic functions with non-homogeneous Neumann boundary conditions, Nonlinear Elliptic Partial Differential Equations, 187–201, Contemp. Math., 540, Amer. Math. Soc., Providence, RI, 2011.
doi: 10.1090/conm/540/10665. |
| [27] |
M. Pérez-Llanos and J. D. Rossi,
The behaviour of the $p(x)$-Laplacian eigenvalue problem as $p(x)\rightarrow \infty$, J. Math. Anal. Appl., 363 (2010), 502-511.
doi: 10.1016/j.jmaa.2009.09.044. |
| [28] |
M. Råžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
| [29] |
V. V. Zhikov,
Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 29 (1987), 33-66.
|
| [30] |
V. V. Zhikov,
On Lavrentiev$'$s phenomenon, Russ. J. Math. Phys., 3 (1995), 249-269.
|
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