June  2020, 40(6): 3171-3200. doi: 10.3934/dcds.2020054

On the spectral theory of positive operators and PDE applications

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

3. 

Department of Mathematics, University of California San Diego, San Diego CA92093, USA

* Corresponding author: Xuefeng Wang

** Current address: School of Science and Engineering, Chinese University of Hong Kong, Shenzhen, Shenzhen 518172, China
Dedicated to the 70th birthday of Professor Wei-Ming Ni

Received  April 2019 Revised  June 2019 Published  October 2019

Fund Project: KCC is supported by NSFC-11371038.
XFW is supported by NSFC-11671190 and NSFC-11731005

The strong version of the Krein-Rutman theorem requires that the positive cone of the Banach space has nonempty interior and the compact map is strongly positive, mapping nonzero points in the cone into its interior. In this paper, we first generalize this version of the Krein-Rutman theorem to the case of "semi-strongly positive" operators; and we prove it in a totally elementary fashion. We then prove the equivalence of semi-strong positivity and irreducibility in a Banach lattice, linking the afore-mentioned result with the Krein-Rutman theorem for irreducible operators. One of the things we emphasize is to use "upper and lower spectral radii" to characterize, in the fashion of Collatz-Wielandt formula for nonnegative irreducible matrices, the principal eigenvalue of these operators. For reducible operators, we prove that the lower spectral radius always serves as the least upper bound of the set of eigenvalues pertaining to positive eigenvectors, and the upper spectral radius the greatest lower bound of the set. Finally, we demonstrate the full power of these Krein-Rutman theorems on some PDE examples such as elliptic eigenvalue problems on non-smooth domains, and cooperative systems which may or may not be fully coupled, by using as few PDE tools as possible.

Citation: Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
References:
[1]

M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk, 19 (1964), 53-161. 

[2]

N. D. Alikakos and G. Fusco, A dynamical systems proof of the Kre$\check{{\rm i }}$n-Rutman theorem and an extension of the Perron theorem, Proc. Roy. Soc. Edinburgh Sect. A, 117 (1991), 209-214.  doi: 10.1017/S0308210500024689.

[3]

J. Barta, Sur la vibration fondamentale d'une membrane, C. R. Acad. Sci. Paris, 204 (1937), 472-473. 

[4]

H. BerestyckiL. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.

[5]

H. BerestyckiF. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0.

[6]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math, 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536.

[7]

I. Birindelli, Hopf's lemma and anti-maximum principle in general domains, J. Diff. Eqs., 119 (1995), 450-472.  doi: 10.1006/jdeq.1995.1098.

[8]

I. BirindelliÉ. Mitidieri and G. Sweers, Existence of the principal eigenvalue for cooperative elliptic systems in a general domain, Differential Equations, 35 (1999), 326-334. 

[9]

K. C. Chang, A nonlinear Krein Rutmann theorem, J. Syst Sci Complex, 22 (2009), 542-544.  doi: 10.1007/s11424-009-9186-2.

[10]

K. C. Chang, Nonlinear extensions of the Perron-Frobenius theorem and the Krein-Rutman theorem, Journal of Fixed Point Theory and Applications, 15 (2014), 433-457.  doi: 10.1007/s11784-014-0191-2.

[11]

L. Collatz, Einschliessungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1946), 221-226.  doi: 10.1007/BF01180013.

[12]

K. Deimling, Nonlinear Functional Analysis, Spronger-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[13]

N. S. Gao, Extensions of Perron-Frobenius theory, Positivity, 17 (2013), 965-977.  doi: 10.1007/s11117-012-0215-3.

[14]

G. Geymonat and P. Grisvard, Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici, Rend. Sem. Mat. Univ. Padova, 38 (1967), 121-173. 

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

J. J. Grobler and C. J. Reinecke, On principal T-bands in a Banach lattice, Integr. equ. oper. theory, 28 (1997), 444-465.  doi: 10.1007/BF01309156.

[17] Z.-Z. Guan, Lectures on functional, Analysis, High Education Press, 1958. 
[18]

R.-J. Jang-Lewis and H. D. Jr Victory, On the ideal structure of positive, eventually compact linear operators on Banach lattices, Pacific Journal of Mathematics, 157 (1993), 57-85.  doi: 10.2140/pjm.1993.157.57.

[19]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[20]

M. G. Kre$\check{{\rm i }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N.S.), 3 (1948), 3-95. 

[21]

K.-Y. Lam and Y. Lou, Asymptotic bahavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Diff Eqs, 28 (2016), 29-48.  doi: 10.1007/s10884-015-9504-4.

[22]

F. Li, J. Coville and X. F. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete and Continuous Dynamical Systems, 37 (2017), 879–903. doi: 10.3934/dcds.2017036.

[23]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.

[24]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.

[25]

F. Niiro and I. Sawashima, On the spectral properties of positive irreducible operators in an arbitrary Banach lattice and problems of H. H. Schaefer, Sci. Papers College Gen. Ed. Univ. Tokyo, 16 (1966), 145-183. 

[26]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[27]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bulll. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.

[28]

M. H. Protter and H. F. Weinberger, Maximum Principles in Diffenretial Equations, Prentice Hall, Inc., Englewood Cliffs, 1967.

[29]

I. Sawashima, On spectral properties of some positive operators, Natur. Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64. 

[30]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[31]

H. H. Schaefer, Some spectral properties of positive linear operators, Pascific J. Math., 10 (1960), 1009-1019.  doi: 10.2140/pjm.1960.10.1009.

[32]

H. H. Schaefer, Topological Vector Spaces, 3rd edition, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.

[33]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Diff. Eqs., 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.

[35]

G. Sweers, Strong positivity in C() for elliptic systems, Math. Z., 209 (1992), 252-271.  doi: 10.1007/BF02570833.

[36]

P. Takáč, A short elementary proof of the Kre$\check{{\rm i }}$n-Rutman Theorem, Houston J. Math., 20 (1994), 93-98. 

[37]

H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications, Trends Math., Birkhäuser/Springer, Cham, (2016), 415–467. doi: 10.1007/978-3-319-27842-1_26.

[38]

W. Walter, A theorem on elliptic differential inequalities and applications to gradient bounds, Math Z., 200 (1989), 293-299.  doi: 10.1007/BF01230289.

[39]

H. Wiedlandt, Unzerlegbare, nicht negative matrizen, Math. Z., 52 (1950), 642-648.  doi: 10.1007/BF02230720.

[40]

E. Zeidler, Nonlinear Functional Analysis and its Applications. IV. Applications to Mathematical Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4566-7.

show all references

References:
[1]

M. S. Agranovič and M. I. Višik, Elliptic problems with a parameter and parabolic problems of general type, Uspehi Mat. Nauk, 19 (1964), 53-161. 

[2]

N. D. Alikakos and G. Fusco, A dynamical systems proof of the Kre$\check{{\rm i }}$n-Rutman theorem and an extension of the Perron theorem, Proc. Roy. Soc. Edinburgh Sect. A, 117 (1991), 209-214.  doi: 10.1017/S0308210500024689.

[3]

J. Barta, Sur la vibration fondamentale d'une membrane, C. R. Acad. Sci. Paris, 204 (1937), 472-473. 

[4]

H. BerestyckiL. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.

[5]

H. BerestyckiF. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Mat. Pura Appl., 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0.

[6]

H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math, 68 (2015), 1014-1065.  doi: 10.1002/cpa.21536.

[7]

I. Birindelli, Hopf's lemma and anti-maximum principle in general domains, J. Diff. Eqs., 119 (1995), 450-472.  doi: 10.1006/jdeq.1995.1098.

[8]

I. BirindelliÉ. Mitidieri and G. Sweers, Existence of the principal eigenvalue for cooperative elliptic systems in a general domain, Differential Equations, 35 (1999), 326-334. 

[9]

K. C. Chang, A nonlinear Krein Rutmann theorem, J. Syst Sci Complex, 22 (2009), 542-544.  doi: 10.1007/s11424-009-9186-2.

[10]

K. C. Chang, Nonlinear extensions of the Perron-Frobenius theorem and the Krein-Rutman theorem, Journal of Fixed Point Theory and Applications, 15 (2014), 433-457.  doi: 10.1007/s11784-014-0191-2.

[11]

L. Collatz, Einschliessungssatz für die charakteristischen Zahlen von Matrizen, Math. Z., 48 (1946), 221-226.  doi: 10.1007/BF01180013.

[12]

K. Deimling, Nonlinear Functional Analysis, Spronger-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[13]

N. S. Gao, Extensions of Perron-Frobenius theory, Positivity, 17 (2013), 965-977.  doi: 10.1007/s11117-012-0215-3.

[14]

G. Geymonat and P. Grisvard, Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici, Rend. Sem. Mat. Univ. Padova, 38 (1967), 121-173. 

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[16]

J. J. Grobler and C. J. Reinecke, On principal T-bands in a Banach lattice, Integr. equ. oper. theory, 28 (1997), 444-465.  doi: 10.1007/BF01309156.

[17] Z.-Z. Guan, Lectures on functional, Analysis, High Education Press, 1958. 
[18]

R.-J. Jang-Lewis and H. D. Jr Victory, On the ideal structure of positive, eventually compact linear operators on Banach lattices, Pacific Journal of Mathematics, 157 (1993), 57-85.  doi: 10.2140/pjm.1993.157.57.

[19]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[20]

M. G. Kre$\check{{\rm i }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N.S.), 3 (1948), 3-95. 

[21]

K.-Y. Lam and Y. Lou, Asymptotic bahavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Diff Eqs, 28 (2016), 29-48.  doi: 10.1007/s10884-015-9504-4.

[22]

F. Li, J. Coville and X. F. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete and Continuous Dynamical Systems, 37 (2017), 879–903. doi: 10.3934/dcds.2017036.

[23]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory Appl., 7 (2010), 103-143.  doi: 10.1007/s11784-010-0010-3.

[24]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.

[25]

F. Niiro and I. Sawashima, On the spectral properties of positive irreducible operators in an arbitrary Banach lattice and problems of H. H. Schaefer, Sci. Papers College Gen. Ed. Univ. Tokyo, 16 (1966), 145-183. 

[26]

B. de Pagter, Irreducible compact operators, Math. Z., 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[27]

M. H. Protter and H. F. Weinberger, On the spectrum of general second order operators, Bulll. Amer. Math. Soc., 72 (1966), 251-255.  doi: 10.1090/S0002-9904-1966-11485-4.

[28]

M. H. Protter and H. F. Weinberger, Maximum Principles in Diffenretial Equations, Prentice Hall, Inc., Englewood Cliffs, 1967.

[29]

I. Sawashima, On spectral properties of some positive operators, Natur. Sci. Rep. Ochanomizu Univ., 15 (1964), 53-64. 

[30]

H. H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York-Heidelberg, 1974.

[31]

H. H. Schaefer, Some spectral properties of positive linear operators, Pascific J. Math., 10 (1960), 1009-1019.  doi: 10.2140/pjm.1960.10.1009.

[32]

H. H. Schaefer, Topological Vector Spaces, 3rd edition, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.

[33]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Diff. Eqs., 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.

[35]

G. Sweers, Strong positivity in C() for elliptic systems, Math. Z., 209 (1992), 252-271.  doi: 10.1007/BF02570833.

[36]

P. Takáč, A short elementary proof of the Kre$\check{{\rm i }}$n-Rutman Theorem, Houston J. Math., 20 (1994), 93-98. 

[37]

H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications, Trends Math., Birkhäuser/Springer, Cham, (2016), 415–467. doi: 10.1007/978-3-319-27842-1_26.

[38]

W. Walter, A theorem on elliptic differential inequalities and applications to gradient bounds, Math Z., 200 (1989), 293-299.  doi: 10.1007/BF01230289.

[39]

H. Wiedlandt, Unzerlegbare, nicht negative matrizen, Math. Z., 52 (1950), 642-648.  doi: 10.1007/BF02230720.

[40]

E. Zeidler, Nonlinear Functional Analysis and its Applications. IV. Applications to Mathematical Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4566-7.

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