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August  2020, 25(8): 3257-3274. doi: 10.3934/dcdsb.2020061

On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

1. 

School of Mathematical Sciences, Inner Mongolia University, Huhhot, 010021, China

2. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

* Corresponding author: Zhiyuan Wen

Received  September 2019 Published  August 2020 Early access  February 2020

Fund Project: The first author is supported the Scientific Starting Research Foundation of Inner Mongolia University (No. 21200-5175108) and the National Natural Science Foundation of China (No. 11901321). The second author is supported by the National Natural Science Foundation of China (No. 11790273)

In this paper, we will first establish the necessary and sufficient conditions for the existence of the principal eigenvalues of second-order measure differential equations with indefinite weighted measures subject to the Neumann boundary condition. Then we will show the principal eigenvalues are continuously dependent on the weighted measures when the weak$^*$ topology is considered for measures. As applications, we will finally solve several optimization problems on principal eigenvalues, including some isospectral problems.

Citation: Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061
References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. Afrouzi and K. Brown, On principle eigenvalues for boundary value problems with indefinite weight and Robin bounary conditions, Proc. Amer. Math. Soc., 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X.

[3]

Y. BenoistP. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.2307/2152750.

[4]

K. Brown and S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[5]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.

[6]

M. Carter and B. Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction, Springer–Verlag, New York, 2000. doi: 10.1007/978-1-4612-1174-7.

[7]

F. CaubetT. Deheuvels and Y. Privat, Optimal location of resources for biased movement of species: The 1D case, SIAM J. Appl. Math., 77 (2017), 1876-1903.  doi: 10.1137/17M1124255.

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differ. Equ., 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333.

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[10]

J. Conway, A Course in Functional Analysis, 2$^{nd}$ edition, Springer–Verlag, New York, 1990.

[11]

J. Eckhardt and A. Kostenko, The inverse spectral problem for indefinite strings, Invent. Math., 204 (2016), 939-977.  doi: 10.1007/s00222-015-0629-1.

[12]

J. Eckhardt and A. Kostenko, The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation,, Int. Math. Res. Not., 176 (2018), 1-18.  doi: 10.1093/imrn/rny176.

[13]

J. EckhardtA. Kostenko and N. Nicolussi, Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow, J. Differ. Equ., 268 (2020), 3016-3034.  doi: 10.1016/j.jde.2019.09.048.

[14]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037–1055, arXiv: math/0605406. doi: 10.4310/PAMQ.2007.v3.n4.a9.

[15]

C. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335.  doi: 10.3934/mbe.2008.5.315.

[16]

Y. Liu, G. Shi and J. Yan, Dependence of solutions and eigenvalues of third order linear measure differential equations on measures, Sci. China Math., 2019. doi: 10.1007/s11425-018-9458-7.

[17]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Math. Biosciences, IV, Lecture Notes in Math., Springer-Verlag, 1922 (2008), 171–205. doi: 10.1007/978-3-540-74331-6_5.

[18]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.

[19]

G. Meng, Extremal problems for eigenvalues of measure differential equations, Proc. Amer. Math. Soc., 143 (2015), 1991-2002.  doi: 10.1090/S0002-9939-2015-12304-0.

[20]

G. Meng, The optimal upper bound for the first eigenvalue of the fourth order equation, Discrete Contin. Dyn. Syst. A, 37 (2017), 6369-6382.  doi: 10.3934/dcds.2017276.

[21]

G. Meng and P. Yan, Optimal lower bound for the first eigenvalue of the fourth order equation, J. Differ. Equ., 261 (2016), 3149-3168.  doi: 10.1016/j.jde.2016.05.018.

[22]

G. MengP. Yan and M. Zhang, Spectrum of one-dimensional $p$-Laplacian with an indefinite integrable weight, Mediterr. J. Math., 7 (2010), 225-248.  doi: 10.1007/s00009-010-0040-5.

[23]

G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differ. Equ., 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001.

[24]

G. Monteiro, A. Slavík and M. Tvrdý, Kurzweil-Stieltjes Integral, Theory and Applications, Ser. Real Anal., Vol. 15, World Scientific, Hackensack, NJ, 2019.

[25]

J. Qi and S. Chen, Extremal norms of the potentials recovered from inverse Dirichlet problems, Inverse Problems, 32 (2016), 035007, 13pp. doi: 10.1088/0266-5611/32/3/035007.

[26]

A. Savchuk and A. Shkalikov, Sturm-Liouville operators with distribution potentials, Trans. Moscow Math. Soc., 64 (2003), 143-192. 

[27]

S. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. doi: 10.1142/1875.

[28]

S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470.  doi: 10.1007/BF01453979.

[29]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis (eds. E.H. Zarantonello and Author 2), Academic Press, (1971), 565–601.

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[31] E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005. 
[32]

A. Teplinsky, Herman's theory revisited, preprint, arXiv: 0707.0078.

[33]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089.  doi: 10.3934/dcdsb.2004.4.1065.

[34]

B. Xie, et al., Extremal problems of the density for vibrating string equations with applications to gap and ratio of eigenvalues, Preprint, 2019.

[35]

M. Zhang, Extremal eigenvalues of measure differential equations with fixed variation, Sci. China Math., 53 (2010), 2573-2588.  doi: 10.1007/s11425-010-4081-9.

[36]

M. Zhang, Minimization of the zeroth Neumann eigenvalues with integrable potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 501-523.  doi: 10.1016/j.anihpc.2012.01.007.

[37]

M. Zhang and Z. Wen, et al., On the number and complete continuity of weighted eigenvalues of measure differential equations, Differ. Integral Equ., 31 (2018), 761-784.

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. Afrouzi and K. Brown, On principle eigenvalues for boundary value problems with indefinite weight and Robin bounary conditions, Proc. Amer. Math. Soc., 127 (1999), 125-130.  doi: 10.1090/S0002-9939-99-04561-X.

[3]

Y. BenoistP. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.2307/2152750.

[4]

K. Brown and S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[5]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003. doi: 10.1002/0470871296.

[6]

M. Carter and B. Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction, Springer–Verlag, New York, 2000. doi: 10.1007/978-1-4612-1174-7.

[7]

F. CaubetT. Deheuvels and Y. Privat, Optimal location of resources for biased movement of species: The 1D case, SIAM J. Appl. Math., 77 (2017), 1876-1903.  doi: 10.1137/17M1124255.

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Differ. Equ., 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333.

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[10]

J. Conway, A Course in Functional Analysis, 2$^{nd}$ edition, Springer–Verlag, New York, 1990.

[11]

J. Eckhardt and A. Kostenko, The inverse spectral problem for indefinite strings, Invent. Math., 204 (2016), 939-977.  doi: 10.1007/s00222-015-0629-1.

[12]

J. Eckhardt and A. Kostenko, The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation,, Int. Math. Res. Not., 176 (2018), 1-18.  doi: 10.1093/imrn/rny176.

[13]

J. EckhardtA. Kostenko and N. Nicolussi, Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow, J. Differ. Equ., 268 (2020), 3016-3034.  doi: 10.1016/j.jde.2019.09.048.

[14]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037–1055, arXiv: math/0605406. doi: 10.4310/PAMQ.2007.v3.n4.a9.

[15]

C. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335.  doi: 10.3934/mbe.2008.5.315.

[16]

Y. Liu, G. Shi and J. Yan, Dependence of solutions and eigenvalues of third order linear measure differential equations on measures, Sci. China Math., 2019. doi: 10.1007/s11425-018-9458-7.

[17]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutorials in Math. Biosciences, IV, Lecture Notes in Math., Springer-Verlag, 1922 (2008), 171–205. doi: 10.1007/978-3-540-74331-6_5.

[18]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.

[19]

G. Meng, Extremal problems for eigenvalues of measure differential equations, Proc. Amer. Math. Soc., 143 (2015), 1991-2002.  doi: 10.1090/S0002-9939-2015-12304-0.

[20]

G. Meng, The optimal upper bound for the first eigenvalue of the fourth order equation, Discrete Contin. Dyn. Syst. A, 37 (2017), 6369-6382.  doi: 10.3934/dcds.2017276.

[21]

G. Meng and P. Yan, Optimal lower bound for the first eigenvalue of the fourth order equation, J. Differ. Equ., 261 (2016), 3149-3168.  doi: 10.1016/j.jde.2016.05.018.

[22]

G. MengP. Yan and M. Zhang, Spectrum of one-dimensional $p$-Laplacian with an indefinite integrable weight, Mediterr. J. Math., 7 (2010), 225-248.  doi: 10.1007/s00009-010-0040-5.

[23]

G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differ. Equ., 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001.

[24]

G. Monteiro, A. Slavík and M. Tvrdý, Kurzweil-Stieltjes Integral, Theory and Applications, Ser. Real Anal., Vol. 15, World Scientific, Hackensack, NJ, 2019.

[25]

J. Qi and S. Chen, Extremal norms of the potentials recovered from inverse Dirichlet problems, Inverse Problems, 32 (2016), 035007, 13pp. doi: 10.1088/0266-5611/32/3/035007.

[26]

A. Savchuk and A. Shkalikov, Sturm-Liouville operators with distribution potentials, Trans. Moscow Math. Soc., 64 (2003), 143-192. 

[27]

S. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. doi: 10.1142/1875.

[28]

S. Senn and P. Hess, On positive solutions of a linear elliptic boundary value problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470.  doi: 10.1007/BF01453979.

[29]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis (eds. E.H. Zarantonello and Author 2), Academic Press, (1971), 565–601.

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[31] E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005. 
[32]

A. Teplinsky, Herman's theory revisited, preprint, arXiv: 0707.0078.

[33]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089.  doi: 10.3934/dcdsb.2004.4.1065.

[34]

B. Xie, et al., Extremal problems of the density for vibrating string equations with applications to gap and ratio of eigenvalues, Preprint, 2019.

[35]

M. Zhang, Extremal eigenvalues of measure differential equations with fixed variation, Sci. China Math., 53 (2010), 2573-2588.  doi: 10.1007/s11425-010-4081-9.

[36]

M. Zhang, Minimization of the zeroth Neumann eigenvalues with integrable potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 501-523.  doi: 10.1016/j.anihpc.2012.01.007.

[37]

M. Zhang and Z. Wen, et al., On the number and complete continuity of weighted eigenvalues of measure differential equations, Differ. Integral Equ., 31 (2018), 761-784.

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