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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  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 & Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061
References:
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R. Adams and J. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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