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A generalized Cox-Ingersoll-Ross equation with growing initial conditions
Vector-valued Schrödinger operators in Lp-spaces
1. | Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany |
2. | Dipartimento di Matematica, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy |
3. | Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy |
In this paper we consider the vector-valued operator div$ (Q\nabla u)-Vu $ of Schrödinger type. Here $ V = (v_{ij}) $ is a nonnegative, locally bounded, matrix-valued function and $ Q $ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $ V $, we assume an that it is pointwise accretive and that its entries are in $ L^\infty_{{\rm loc}}( \mathbb{R}^d) $. Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a $ C_0 $-semigroup of contractions in $ L^p( \mathbb{R}^d; \mathbb{C}^m) $. Further properties are also investigated.
References:
[1] |
D. Addona, L. Angiuli, L. Lorenzi and G. Tessitore,
On coupled systems of Kolmogorov equations with applications to stochastic differential games, ESAIM Control, Optim. Calc. of Var., 23 (2017), 937-976.
doi: 10.1051/cocv/2016019. |
[2] |
S. Agmon,
The $L_{p}$ approach to the Dirichlet problem. Ⅰ. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 405-448.
|
[3] |
L. Angiuli, L. Lorenzi and D. Pallara,
$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order, J. Math. Anal. Appl., 444 (2016), 110-135.
doi: 10.1016/j.jmaa.2016.06.001. |
[4] |
V. Betz, B. D. Goddard and S. Teufel,
Superadiabatic transitions in quantum molecular dynamics, Proc. R. Soc. A, 465 (2009), 3553-3580.
doi: 10.1098/rspa.2009.0337. |
[5] |
G. M. Dall'Ara,
Discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, J. Funct. Anal., 268 (2015), 3649-3679.
doi: 10.1016/j.jfa.2014.10.007. |
[6] |
S. Delmonte and L. Lorenzi,
On a class of weakly coupled systems of elliptic operators with unbounded coefficients, Milan J. Math., 79 (2011), 689-727.
doi: 10.1007/s00032-011-0170-7. |
[7] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[8] |
S. Fornaro and L. Lorenzi,
Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discr. Cont. Dyn. Syst., 18 (2007), 747-772.
doi: 10.3934/dcds.2007.18.747. |
[9] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin, 2001. |
[10] |
T. Hansel and A. Rhandi,
The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case, J. Rein. Angew. Math., 694 (2014), 1-26.
doi: 10.1515/crelle-2012-0113. |
[11] |
F. Haslinger and B. Helffer,
Compactness of the solution operator to $\overline{\partial}$ in weighted $L^2$-spaces, J. Funct. Anal., 243 (2007), 679-697.
doi: 10.1016/j.jfa.2006.09.004. |
[12] |
M. Hieber, L. Lorenzi, J. Prüss and A. Rhandi,
Global properties of generalized Ornstein-Uhlenbeck operators on $L^p(\mathbb{R}^N, \mathbb{R}^N)$ with more than linearly growing coefficients, J. Math. Anal. Appl., 350 (2009), 100-121.
doi: 10.1016/j.jmaa.2008.09.011. |
[13] |
M. Hieber, A. Rhandi and O. Sawada,
The Navier-Stokes flow for globally Lipschitz continuous initial data, Kyoto Conference on the Navier-Stokes Equations and their Applications, Res. Inst. Math. Sci. (RIMS) Kkyroku Bessatsu, B1 (2007), 159-165.
|
[14] |
M. Hieber and O. Sawada,
The Navier-Stokes equations in $\mathbb{R}^n$ with linearly growing initial data, Arch. Ration. Mech. Anal., 175 (2005), 269-285.
doi: 10.1007/s00205-004-0347-0. |
[15] |
T. Kato,
On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 105-114.
|
[16] |
M. Kunze, L. Lorenzi, A. Maichine and A. Rhandi, $L^p$-theory for Schrödinger systems, to appear in Math. Nachr, doi: 10.1002/mana.201800206, 2019.
doi: 10.1002/mana.201800206. |
[17] |
L. Lorenzi, Analytical Methods for Kolmogorov Equations, Second edition, Monograph and research notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2017. |
[18] |
L. Lorenzi and A. Rhandi,
On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.
doi: 10.1007/s00028-014-0249-z. |
[19] |
A. Maichine and A. Rhandi,
On a polynomial scalar perturbation of a Schrödinger system in $L^p$-spaces, J. Math. Anal. Appl., 466 (2018), 655-675.
doi: 10.1016/j.jmaa.2018.06.014. |
[20] |
J. Prüss, A. Rhandi and R. Schnaubelt,
The domain of elliptic operators on $L^p(\mathbb{R}^d)$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.
|
[21] |
K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1980. |
show all references
References:
[1] |
D. Addona, L. Angiuli, L. Lorenzi and G. Tessitore,
On coupled systems of Kolmogorov equations with applications to stochastic differential games, ESAIM Control, Optim. Calc. of Var., 23 (2017), 937-976.
doi: 10.1051/cocv/2016019. |
[2] |
S. Agmon,
The $L_{p}$ approach to the Dirichlet problem. Ⅰ. Regularity theorems, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 405-448.
|
[3] |
L. Angiuli, L. Lorenzi and D. Pallara,
$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order, J. Math. Anal. Appl., 444 (2016), 110-135.
doi: 10.1016/j.jmaa.2016.06.001. |
[4] |
V. Betz, B. D. Goddard and S. Teufel,
Superadiabatic transitions in quantum molecular dynamics, Proc. R. Soc. A, 465 (2009), 3553-3580.
doi: 10.1098/rspa.2009.0337. |
[5] |
G. M. Dall'Ara,
Discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, J. Funct. Anal., 268 (2015), 3649-3679.
doi: 10.1016/j.jfa.2014.10.007. |
[6] |
S. Delmonte and L. Lorenzi,
On a class of weakly coupled systems of elliptic operators with unbounded coefficients, Milan J. Math., 79 (2011), 689-727.
doi: 10.1007/s00032-011-0170-7. |
[7] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[8] |
S. Fornaro and L. Lorenzi,
Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discr. Cont. Dyn. Syst., 18 (2007), 747-772.
doi: 10.3934/dcds.2007.18.747. |
[9] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin, 2001. |
[10] |
T. Hansel and A. Rhandi,
The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The non-autonomous case, J. Rein. Angew. Math., 694 (2014), 1-26.
doi: 10.1515/crelle-2012-0113. |
[11] |
F. Haslinger and B. Helffer,
Compactness of the solution operator to $\overline{\partial}$ in weighted $L^2$-spaces, J. Funct. Anal., 243 (2007), 679-697.
doi: 10.1016/j.jfa.2006.09.004. |
[12] |
M. Hieber, L. Lorenzi, J. Prüss and A. Rhandi,
Global properties of generalized Ornstein-Uhlenbeck operators on $L^p(\mathbb{R}^N, \mathbb{R}^N)$ with more than linearly growing coefficients, J. Math. Anal. Appl., 350 (2009), 100-121.
doi: 10.1016/j.jmaa.2008.09.011. |
[13] |
M. Hieber, A. Rhandi and O. Sawada,
The Navier-Stokes flow for globally Lipschitz continuous initial data, Kyoto Conference on the Navier-Stokes Equations and their Applications, Res. Inst. Math. Sci. (RIMS) Kkyroku Bessatsu, B1 (2007), 159-165.
|
[14] |
M. Hieber and O. Sawada,
The Navier-Stokes equations in $\mathbb{R}^n$ with linearly growing initial data, Arch. Ration. Mech. Anal., 175 (2005), 269-285.
doi: 10.1007/s00205-004-0347-0. |
[15] |
T. Kato,
On some Schrödinger operators with a singular complex potential, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 105-114.
|
[16] |
M. Kunze, L. Lorenzi, A. Maichine and A. Rhandi, $L^p$-theory for Schrödinger systems, to appear in Math. Nachr, doi: 10.1002/mana.201800206, 2019.
doi: 10.1002/mana.201800206. |
[17] |
L. Lorenzi, Analytical Methods for Kolmogorov Equations, Second edition, Monograph and research notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2017. |
[18] |
L. Lorenzi and A. Rhandi,
On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.
doi: 10.1007/s00028-014-0249-z. |
[19] |
A. Maichine and A. Rhandi,
On a polynomial scalar perturbation of a Schrödinger system in $L^p$-spaces, J. Math. Anal. Appl., 466 (2018), 655-675.
doi: 10.1016/j.jmaa.2018.06.014. |
[20] |
J. Prüss, A. Rhandi and R. Schnaubelt,
The domain of elliptic operators on $L^p(\mathbb{R}^d)$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.
|
[21] |
K. Yosida, Functional Analysis, Springer-Verlag, Berlin Heidelberg, New York, 1980. |
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