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June  2011, 4(3): 595-614. doi: 10.3934/dcdss.2011.4.595

Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces

Angelo Favini 1, and  Yakov Yakubov 2,

1. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna
2. 

Raymond and Beverly Sackler School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Received  December 2008 Revised  November 2009 Published  November 2010

We prove an isomorphism of nonlocal boundary value problems for higher order ordinary differential-operator equations generated by one operator in UMD Banach spaces in appropriate Sobolev and interpolation spaces. The main condition is given in terms of $\R$-boundedness of some families of bounded operators generated by the resolvent of the operator of the equation. This implies maximal $L_p$-regularity for the problem. Then we study Fredholmnees of more general problems, namely, with linear abstract perturbation operators both in the equation and boundary conditions. We also present an application of obtained abstract results to boundary value problems for higher order elliptic partial differential equations.
Keywords: UMD Banach space, $\R$-boundedness, Fredholmness, higher order., isomorphism, regular problem, Abstract elliptic equation.
Mathematics Subject Classification: Primary: 34G10, 47E05; Secondary: 35J40, 47N2.
Citation: Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II, Comm. Pure Appl. Math., 12 (1959), 623-727; 17 (1964), 35-92.  Google Scholar

[2]

W. Arendt and M. Duelli, Maximal $L^p$-regularity for parabolic and elliptic equations on the line, J. Evol. Equ., 6 (2006), 773-790. doi: doi:10.1007/s00028-006-0292-5.  Google Scholar

[3]

W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.  Google Scholar

[4]

R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni, New thoughts on old results of R. T. Seeley, Mathematische Annalen, 328 (2004), 545-583. doi: doi:10.1007/s00208-003-0493-y.  Google Scholar

[5]

R. Denk, M. Hieber and J. Prüss, "$R$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type," Mem. Amer. Math. Soc., Providence, 2003.  Google Scholar

[6]

A. Favini, V. Shakhmurov and Ya. Yakubov, Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum, 79 (2009), 22-54. doi: doi:10.1007/s00233-009-9138-0.  Google Scholar

[7]

A. Favini and Ya. Yakubov, Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces, Differential and Integral Equations, 21 (2008), 497-512.  Google Scholar

[8]

A. Favini and Ya. Yakubov, Regular boundary value problems for elliptic differential-operator equations of the fourth order in UMD Banach spaces, Scientiae Mathematicae Japonicae, 70 (2009), 183-204. Google Scholar

[9]

A. Favini and Ya. Yakubov, Irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces, Mathematische Annalen, 348 (2010), 601-632. doi: doi:10.1007/s00208-010-0491-9.  Google Scholar

[10]

N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Mathematische Annalen, 336 (2006), 747-801. doi: doi:10.1007/s00208-005-0742-3.  Google Scholar

[11]

N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Mathematische Annalen, 321 (2001), 319-345. doi: doi:10.1007/s002080100231.  Google Scholar

[12]

P. C. Kunstmann and L. Weis, "Maximal $L_p$-Regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$-Functional Calculus," in "Functional Analytic Methods for Evolution Equations," Lecture Notes in Mathematics, 1855, Springer, (2004), 65-311.  Google Scholar

[13]

H. Triebel, "Interpolation Theory. Function Spaces. Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[14]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Mathematische Annalen, 319 (2001), 735-758. doi: doi:10.1007/PL00004457.  Google Scholar

[15]

S. Yakubov and Ya. Yakubov, "Differential-Operator Equations. Ordinary and Partial Differential Equations," Chapman and Hall/CRC, Boca Raton, 2000.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, II, Comm. Pure Appl. Math., 12 (1959), 623-727; 17 (1964), 35-92.  Google Scholar

[2]

W. Arendt and M. Duelli, Maximal $L^p$-regularity for parabolic and elliptic equations on the line, J. Evol. Equ., 6 (2006), 773-790. doi: doi:10.1007/s00028-006-0292-5.  Google Scholar

[3]

W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.  Google Scholar

[4]

R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni, New thoughts on old results of R. T. Seeley, Mathematische Annalen, 328 (2004), 545-583. doi: doi:10.1007/s00208-003-0493-y.  Google Scholar

[5]

R. Denk, M. Hieber and J. Prüss, "$R$-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type," Mem. Amer. Math. Soc., Providence, 2003.  Google Scholar

[6]

A. Favini, V. Shakhmurov and Ya. Yakubov, Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum, 79 (2009), 22-54. doi: doi:10.1007/s00233-009-9138-0.  Google Scholar

[7]

A. Favini and Ya. Yakubov, Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces, Differential and Integral Equations, 21 (2008), 497-512.  Google Scholar

[8]

A. Favini and Ya. Yakubov, Regular boundary value problems for elliptic differential-operator equations of the fourth order in UMD Banach spaces, Scientiae Mathematicae Japonicae, 70 (2009), 183-204. Google Scholar

[9]

A. Favini and Ya. Yakubov, Irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces, Mathematische Annalen, 348 (2010), 601-632. doi: doi:10.1007/s00208-010-0491-9.  Google Scholar

[10]

N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Mathematische Annalen, 336 (2006), 747-801. doi: doi:10.1007/s00208-005-0742-3.  Google Scholar

[11]

N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Mathematische Annalen, 321 (2001), 319-345. doi: doi:10.1007/s002080100231.  Google Scholar

[12]

P. C. Kunstmann and L. Weis, "Maximal $L_p$-Regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$-Functional Calculus," in "Functional Analytic Methods for Evolution Equations," Lecture Notes in Mathematics, 1855, Springer, (2004), 65-311.  Google Scholar

[13]

H. Triebel, "Interpolation Theory. Function Spaces. Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[14]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Mathematische Annalen, 319 (2001), 735-758. doi: doi:10.1007/PL00004457.  Google Scholar

[15]

S. Yakubov and Ya. Yakubov, "Differential-Operator Equations. Ordinary and Partial Differential Equations," Chapman and Hall/CRC, Boca Raton, 2000.  Google Scholar

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