抽象边值问题与算子半群
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摘要
本文共分三章。第一章首先证明了关于Hille-Yosida算子的两种无界扰动仍是Hille-Yosida算子的两个扰动定理，然后依此给出了边界扰动抽象边值问题的适定性的两种判别方法。第二章利用算子矩阵的分解分别给出了边界算子无界和有界两种情形下抽象动态边值问题解析性的判别方法。第三章利用算子矩阵和正半群的结果给出了抽象动态边值问题的正性和稳定性的等价刻画，推广了文[5]的结果，作为应用，讨论了时滞微分方程的正性和稳定性。
This thesis is divided into three parts. In the first character,two theorems about the unbounded perturbation of Hille — Yosida operator are proved,and with them we give two results on how to characterise the wellposedness of abstract boundary value problem with boundary perturbation.In the second character,we characterise the analyticity of abstract dynamic boundary value problem with unbounded boundary operator and abstract dynamic boundary value problem with bounded boundary operator using the decomposition of operator matrices.In the last character,the necessary and sufficient conditions for the positivity and stability of abstract dynamic boundary value problem are presented,which generalise some results in [5].As application,we discuss the positivity and stability of differential equation with delay.
This thesis is divided into three parts. In the first character,two theorems about the unbounded perturbation of Hille — Yosida operator are proved,and with them we give two results on how to characterise the wellposedness of abstract boundary value problem with boundary perturbation.In the second character,we characterise the analyticity of abstract dynamic boundary value problem with unbounded boundary operator and abstract dynamic boundary value problem with bounded boundary operator using the decomposition of operator matrices.In the last character,the necessary and sufficient conditions for the positivity and stability of abstract dynamic boundary value problem are presented,which generalise some results in [5].As application,we discuss the positivity and stability of differential equation with delay.
引文
[1] K, -J, Engel: Matrix methods for Wentzell boundary conditions. Preprint.
[2] K, -J, Engel: Positivity and stability for one-sided coupled operator matrices. Positivity 1(1997), 103-124.
[3] K, -J, Engel: Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum 58 (1999), 267-295.
[4] G. Nickel: A new look at boundary perturbations of generators. Tiibinger Berichte zur Funktionalanalysis 12, 2002.
[5] V. Casarino, K. Engel, R. Nagel, G. Nickel: A semigroup approach to boundary feedback systems. Integral Equations Operator Theory 47(2003), 289-306.
[6] G. Nickel: A semigroup approach to dynamic boundary value problems. Semigroup Forum. 69(2004), 159-183.
[7] G. Nickel: A semigroup approach to dynamic boundary value problems with bounded boundary operator. Tiibinger Berichte zur Funktionalanalysis 12, 2002.
[8] G. Greiner: Perturbing the boundary conditions of a generator. Houston J. Math. 13(1987), 213-229.
[9] T. -J. Xiao and J. Liang: Wave equations with Wentzell boundary conditions. Math. Ann. 327(2003), 351-363.
[10] G. Greiner, K. Kuhu: Linear and semilinear boundary conditions: The analytic case. Lectares notes in Pure and Applied Mathematics 135. Marcel Dekker (1991)193-211.
[11] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander: Vector-valued Laplace Transform and Cauthy Problems. Monographs Maths, Vol 96, Birkhiiuser Verleg, 2001.
[12] K. Engel, R. Nagel: One-Parameter semigroups for linear Evolution Equations. GTM194. Springer-Verlag(2000).
[13] R. Nagel: One-Parameter Semigroups of Positivity Operators, Lect. Notes in Math, 1184. Springer-Verlag(1986).
[14] Angus E. Taylor, Daud L. Lay: Introduction to Functional Analysis. second edition, Wiley(1980).
[15] A. Pazy: Semigroups of linear Operators and Applications to Partial Differential Equations. Springer-Verlag(1983).
[16] R.Nagel:Towards a "matrix theory" for unbounded oerator matrices.Math.Z.201(1989),57- 68.
[17] A.Rhandi.Positivity and stability for a population equation with diffudion in L~1 .Positivity 2(1998),101-113.
[18] R.NagelrCharacteristic equations for the spectrum of generators. Ann.Sc.Norm.Sup.Pisa. 24(1997),703-717.
[19] Said Hadd:Unbounded Perturbations of C-0 semigroups on Banach Spaces and Appli- cation.Semigroup Forum. 70(2005),451-465.
[20] L.Lasiecka,R.Triggani: " Control Theory for Partial Differential Equations:Continuous and Approximation Theories ". Cambridge University Press,2000.
[21] G.Nickel ,A.Rhandi:On the essential spectral radius of semigroups generated by pertur- bations of Hille-Yosida operations.Tubinger Berichte zur Punktionalanalysis 4(2002) ,207- 220.
[2] K, -J, Engel: Positivity and stability for one-sided coupled operator matrices. Positivity 1(1997), 103-124.
[3] K, -J, Engel: Spectral theory and generator property for one-sided coupled operator matrices. Semigroup Forum 58 (1999), 267-295.
[4] G. Nickel: A new look at boundary perturbations of generators. Tiibinger Berichte zur Funktionalanalysis 12, 2002.
[5] V. Casarino, K. Engel, R. Nagel, G. Nickel: A semigroup approach to boundary feedback systems. Integral Equations Operator Theory 47(2003), 289-306.
[6] G. Nickel: A semigroup approach to dynamic boundary value problems. Semigroup Forum. 69(2004), 159-183.
[7] G. Nickel: A semigroup approach to dynamic boundary value problems with bounded boundary operator. Tiibinger Berichte zur Funktionalanalysis 12, 2002.
[8] G. Greiner: Perturbing the boundary conditions of a generator. Houston J. Math. 13(1987), 213-229.
[9] T. -J. Xiao and J. Liang: Wave equations with Wentzell boundary conditions. Math. Ann. 327(2003), 351-363.
[10] G. Greiner, K. Kuhu: Linear and semilinear boundary conditions: The analytic case. Lectares notes in Pure and Applied Mathematics 135. Marcel Dekker (1991)193-211.
[11] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander: Vector-valued Laplace Transform and Cauthy Problems. Monographs Maths, Vol 96, Birkhiiuser Verleg, 2001.
[12] K. Engel, R. Nagel: One-Parameter semigroups for linear Evolution Equations. GTM194. Springer-Verlag(2000).
[13] R. Nagel: One-Parameter Semigroups of Positivity Operators, Lect. Notes in Math, 1184. Springer-Verlag(1986).
[14] Angus E. Taylor, Daud L. Lay: Introduction to Functional Analysis. second edition, Wiley(1980).
[15] A. Pazy: Semigroups of linear Operators and Applications to Partial Differential Equations. Springer-Verlag(1983).
[16] R.Nagel:Towards a "matrix theory" for unbounded oerator matrices.Math.Z.201(1989),57- 68.
[17] A.Rhandi.Positivity and stability for a population equation with diffudion in L~1 .Positivity 2(1998),101-113.
[18] R.NagelrCharacteristic equations for the spectrum of generators. Ann.Sc.Norm.Sup.Pisa. 24(1997),703-717.
[19] Said Hadd:Unbounded Perturbations of C-0 semigroups on Banach Spaces and Appli- cation.Semigroup Forum. 70(2005),451-465.
[20] L.Lasiecka,R.Triggani: " Control Theory for Partial Differential Equations:Continuous and Approximation Theories ". Cambridge University Press,2000.
[21] G.Nickel ,A.Rhandi:On the essential spectral radius of semigroups generated by pertur- bations of Hille-Yosida operations.Tubinger Berichte zur Punktionalanalysis 4(2002) ,207- 220.