March  2022, 42(3): 1415-1434. doi: 10.3934/dcds.2021158

Lower bounds for Orlicz eigenvalues

Instituto de Cálculo, FCEyN - Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Av. Cantilo s/n., Buenos Aires, Argentina

Received  April 2021 Revised  September 2021 Published  March 2022 Early access  November 2021

In this article we consider the following weighted nonlinear eigenvalue problem for the
$ g- $
Laplacian
$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $
with Dirichlet boundary conditions. Here
$ w $
is a suitable weight and
$ g = G' $
and
$ h = H' $
are appropriated Young functions satisfying the so called
$ \Delta' $
condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of
$ G $
,
$ H $
,
$ w $
and the normalization
$ \mu $
of the corresponding eigenfunctions.
We introduce some new strategies to obtain results that generalize several inequalities from the literature of
$ p- $
Laplacian type eigenvalues.
Citation: Ariel Salort. Lower bounds for Orlicz eigenvalues. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1415-1434. doi: 10.3934/dcds.2021158
References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. Anane, Simplicité et isolation de la premiere valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728. 

[3]

S. Bahrouni and A. Salort, Neumann and Robin type boundary conditions in fractional Orlicz-Sobolev spaces, ESAIM Control Optim. Calc. Var., 27 (2021), 23pp. doi: 10.1051/cocv/2020064.

[4]

A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc., 351 (1999), 2459-2478.  doi: 10.1090/S0002-9947-99-01985-6.

[5]

M. Cuesta, Eigenvalue problems for the-Laplacian with indefinite weights, Electron. J. Differential Equations, 2001 (2001), 9pp.

[6]

J. V. Da SilvaA. SalortA. Silva and J. Spedaletti, A constrained shape optimization problem in Orlicz-Sobolev spaces, J. Differential Equations, 267 (2019), 5493-5520.  doi: 10.1016/j.jde.2019.05.038.

[7]

P. De Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018.  doi: 10.1016/j.jfa.2016.01.006.

[8]

P. De Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200. 

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai, 30 (1979), 153-180. 

[11]

J. Fernández Bonder and A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.

[12]

M. García-HuidobroV. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.

[13]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.

[14]

J.-P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909.  doi: 10.1017/S030821050000192X.

[15]

M. JleliM. Kirane and B. Samet, Lyapunov-type inequalities for fractional partial differential equations, Appl. Math. Lett., 66 (2017), 30-39.  doi: 10.1016/j.aml.2016.10.013.

[16]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen, 1961.

[17]

A. Kufner, O. John and S. Fučík, Function Spaces, Springer Science & Business Media, 1977.

[18]

C. LeeC. YehC. Hong and R. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853.  doi: 10.1016/j.aml.2004.06.016.

[19]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uralltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[20]

A. Liapounoff, Probléme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203-474. 

[21]

M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ., 19 (1984), 23-28. 

[22]

B. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148.  doi: 10.1515/GMJ.1997.139.

[23]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional $p$-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153.  doi: 10.1155/S108533750431002X.

[24]

A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations, 268 (2020), 5413-5439.  doi: 10.1016/j.jde.2019.11.027.

[25]

A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition, preprint, 2020, arXiv: 2005.01847.

[26]

J. Sánchez and V. Vergara, A Lyapunov-type inequality for a $\phi$-Laplacian operator, Nonlinear Anal.: Theory, Methods and Applications, 74 (2011), 7071-7077.  doi: 10.1016/j.na.2011.07.027.

[27]

I. Sim and Y. Lee, Lyapunov inequalities for one-dimensional-Laplacian problems with a singular weight function, J. Inequal. Appl., 2010 (2010), 1-9.  doi: 10.1155/2010/865096.

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. Anane, Simplicité et isolation de la premiere valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 725-728. 

[3]

S. Bahrouni and A. Salort, Neumann and Robin type boundary conditions in fractional Orlicz-Sobolev spaces, ESAIM Control Optim. Calc. Var., 27 (2021), 23pp. doi: 10.1051/cocv/2020064.

[4]

A. Cianchi, Hardy inequalities in Orlicz spaces, Trans. Amer. Math. Soc., 351 (1999), 2459-2478.  doi: 10.1090/S0002-9947-99-01985-6.

[5]

M. Cuesta, Eigenvalue problems for the-Laplacian with indefinite weights, Electron. J. Differential Equations, 2001 (2001), 9pp.

[6]

J. V. Da SilvaA. SalortA. Silva and J. Spedaletti, A constrained shape optimization problem in Orlicz-Sobolev spaces, J. Differential Equations, 267 (2019), 5493-5520.  doi: 10.1016/j.jde.2019.05.038.

[7]

P. De Nápoli and J. P. Pinasco, Lyapunov-type inequalities for partial differential equations, J. Funct. Anal., 270 (2016), 1995-2018.  doi: 10.1016/j.jfa.2016.01.006.

[8]

P. De Nápoli and J. P. Pinasco, A Lyapunov inequality for monotone quasilinear operators, Differential Integral Equations, 18 (2005), 1193-1200. 

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai, 30 (1979), 153-180. 

[11]

J. Fernández Bonder and A. Salort, Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., 277 (2019), 333-367.  doi: 10.1016/j.jfa.2019.04.003.

[12]

M. García-HuidobroV. LeR. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 207-225.  doi: 10.1007/s000300050073.

[13]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.1090/S0002-9947-1974-0342854-2.

[14]

J.-P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 891-909.  doi: 10.1017/S030821050000192X.

[15]

M. JleliM. Kirane and B. Samet, Lyapunov-type inequalities for fractional partial differential equations, Appl. Math. Lett., 66 (2017), 30-39.  doi: 10.1016/j.aml.2016.10.013.

[16]

M. A. Krasnosel'skiĭ and J. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff Ltd., Groningen, 1961.

[17]

A. Kufner, O. John and S. Fučík, Function Spaces, Springer Science & Business Media, 1977.

[18]

C. LeeC. YehC. Hong and R. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847-853.  doi: 10.1016/j.aml.2004.06.016.

[19]

G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uralltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[20]

A. Liapounoff, Probléme général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1907), 203-474. 

[21]

M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ., 19 (1984), 23-28. 

[22]

B. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J., 4 (1997), 139-148.  doi: 10.1515/GMJ.1997.139.

[23]

J. P. Pinasco, Lower bounds for eigenvalues of the one-dimensional $p$-Laplacian, Abstr. Appl. Anal., 2004 (2004), 147-153.  doi: 10.1155/S108533750431002X.

[24]

A. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations, 268 (2020), 5413-5439.  doi: 10.1016/j.jde.2019.11.027.

[25]

A. Salort and H. Vivas, Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition, preprint, 2020, arXiv: 2005.01847.

[26]

J. Sánchez and V. Vergara, A Lyapunov-type inequality for a $\phi$-Laplacian operator, Nonlinear Anal.: Theory, Methods and Applications, 74 (2011), 7071-7077.  doi: 10.1016/j.na.2011.07.027.

[27]

I. Sim and Y. Lee, Lyapunov inequalities for one-dimensional-Laplacian problems with a singular weight function, J. Inequal. Appl., 2010 (2010), 1-9.  doi: 10.1155/2010/865096.

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