• Previous Article
    Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model
  • DCDS Home
  • This Issue
  • Next Article
    Preface special issue: Advances and applications in qualitative studies of dynamics
February  2015, 35(2): 595-615. doi: 10.3934/dcds.2015.35.595

Analytic semigroups and some degenerate evolution equations defined on domains with corners

1. 

Dipartimento di Matematica e Fisica “E. De Giorgi", Università del Salento, Via Per Arnesano, P.O. Box 193, I-73100 Lecce, Italy

2. 

Department of Mathematics “E. De Giorgi”, University of Salento, P.O. Box 193, Via Per Arnesano, 73100 Lecce, Italy

Received  January 2013 Revised  November 2013 Published  September 2014

We study the analyticity of the semigroups generated by some classes of degenerate second order differential operators in the space of continuous function on a domain with corners. These semigroups arise from the theory of dynamics of populations.
Citation: Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595
References:
[1]

A. A. Albanese, M. Campiti and E. Mangino, Approximation formulas for $C_0$-semigroups and their resolvent, J. Appl. Funct. Anal., 1 (2006), 343-358.

[2]

A. A. Albanese, M. Campiti and E. Mangino, Regularity properties of semigroups generated by some Fleming-Viot type operators, J. Math. Anal. Appl., 335 (2007), 1259-1273. doi: 10.1016/j.jmaa.2007.02.042.

[3]

A. A. Albanese and E. Mangino, A class of non-symmetric forms on the canonical simplex of $S^d$, Discrete and Continuous Dynamical Systems-Series A, 23 (2009), 639-654. doi: 10.3934/dcds.2009.23.639.

[4]

A. A. Albanese and E. Mangino, Analyticity of a class of degenerate evolution equations on the simplex of $S^d$ arising from Fleming-Viot processes, J. Math. Anal. Appl., 379 (2011), 401-424. doi: 10.1016/j.jmaa.2011.01.015.

[5]

A. A. Albanese and E. Mangino, One-dimensional degenerate diffusion operators, Mediterr. J. Math., 10 (2013), 707-729. doi: 10.1007/s00009-013-0279-8.

[6]

S. Angenent, Local existence and regularity for a class of degenerate parabolic equations, Math. Ann., 280 (1988), 465-482. doi: 10.1007/BF01456337.

[7]

S. R. Athreya, R. F. Bass and E. A. Perkins, Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces, Trans. Amer. Math. Soc., 357 (2005), 5001-5029. doi: 10.1090/S0002-9947-05-03638-X.

[8]

R. F. Bass and E. A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc., 355 (2002), 373-405. doi: 10.1090/S0002-9947-02-03120-3.

[9]

H. Brezis, W. Rosenkrants and B. Singer, On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math., 24 (1971), 395-416. doi: 10.1002/cpa.3160240305.

[10]

M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390. doi: 10.1007/s000130050210.

[11]

M. Campiti and I. Rasa, Qualitative properties of a class of Fleming-Viot operators, Acta Math. Hungar., 103 (2004), 55-69. doi: 10.1023/B:AMHU.0000028236.59446.da.

[12]

S. Cerrai and P. Clément, On a class of degenerate elliptic operators arising from the Fleming-Viot processes, J. Evol. Equ., 1 (2001), 243-276. doi: 10.1007/PL00001370.

[13]

S. Cerrai and P. Clément, Schauder estimates for a degenerate second-order elliptic operator on a cube, J. Differential Equations, 242 (2007), 287-321. doi: 10.1016/j.jde.2007.08.002.

[14]

P. Clément and C. A. Timmermans, On $C_0$-semigroup generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math., 89 (1986), 379-387.

[15]

J. R. Dorroh, Contraction semi-groups in a function space, Pacific J. Math., 19 (1966), 35-38. doi: 10.2140/pjm.1966.19.35.

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, Berlin, Heildelberg, 2000.

[17]

C. L. Epstein and R. Mazzeo, Wright-Fisher diffusion in one dimension, SIAM J. Math. Anal., 42 (2010), 1429-1436. doi: 10.1137/090766152.

[18]

C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Annals of Math. Studies, Princeton University Press, 2012.

[19]

S. N. Ethier, A class of degenerate diffusion processes occurring in population genetics, Comm. Pure Appl. Math., 29 (1976), 483-493. doi: 10.1002/cpa.3160290503.

[20]

S. N. Ethier and T. G. Kurtz, Markov Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1986. doi: 10.1002/9780470316658.

[21]

S. N. Ethier and T. G. Kurtz, Fleming-Viot processes in population genetics, SIAM J. Control Optim., 31 (1993), 345-386. doi: 10.1137/0331019.

[22]

W. Feller, Two singular diffusion problems, Ann. of Math., 54 (1951), 173-181. doi: 10.2307/1969318.

[23]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. doi: 10.2307/1969644.

[24]

W. H. Fleming and M. Viot, Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J., 28 (1979), 817-843. doi: 10.1512/iumj.1979.28.28058.

[25]

H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1980.

[26]

G. Köthe, Topological Vector Spaces II, Springer Verlag, Berlin-Heidelberg-New York, 1979.

[27]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[28]

G. Metafune, Analiticity for some degenerate one-dimensional evolution equations, Studia Math., 127 (1998), 251-276.

[29]

R. Nagel, One-Parameter Semigroups of Positive Operators, Lect. Notes Math., 1184, Springer, 1986.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

S. Pal, Analysis of the market weights under the volatility-stabilized market mode, Ann. App. Prob., 21 (2011), 1180-2013. doi: 10.1214/10-AAP725.

[32]

N. Shimakura, Equations différentielles provenant de la génétique des populations, Tôhoku Math. J., 77 (1977), 287-318.

[33]

N. Shimakura, Formulas for diffusion approximations of some gene frequency models, J. Math. Kyoto Univ., 21 (1981), 19-45.

[34]

N. Shimakura, Partial Differential Operators of Elliptic Type, Translations of Mathematical Monographs, 99, Amer. Math. Soc., Providence, 1992.

[35]

W. Stannat, On the validity of the logarithmic-Sobolev inequality for symmetric Fleming-Viot operators, Annals Prob., 28 (2000), 667-684. doi: 10.1214/aop/1019160256.

[36]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.

show all references

References:
[1]

A. A. Albanese, M. Campiti and E. Mangino, Approximation formulas for $C_0$-semigroups and their resolvent, J. Appl. Funct. Anal., 1 (2006), 343-358.

[2]

A. A. Albanese, M. Campiti and E. Mangino, Regularity properties of semigroups generated by some Fleming-Viot type operators, J. Math. Anal. Appl., 335 (2007), 1259-1273. doi: 10.1016/j.jmaa.2007.02.042.

[3]

A. A. Albanese and E. Mangino, A class of non-symmetric forms on the canonical simplex of $S^d$, Discrete and Continuous Dynamical Systems-Series A, 23 (2009), 639-654. doi: 10.3934/dcds.2009.23.639.

[4]

A. A. Albanese and E. Mangino, Analyticity of a class of degenerate evolution equations on the simplex of $S^d$ arising from Fleming-Viot processes, J. Math. Anal. Appl., 379 (2011), 401-424. doi: 10.1016/j.jmaa.2011.01.015.

[5]

A. A. Albanese and E. Mangino, One-dimensional degenerate diffusion operators, Mediterr. J. Math., 10 (2013), 707-729. doi: 10.1007/s00009-013-0279-8.

[6]

S. Angenent, Local existence and regularity for a class of degenerate parabolic equations, Math. Ann., 280 (1988), 465-482. doi: 10.1007/BF01456337.

[7]

S. R. Athreya, R. F. Bass and E. A. Perkins, Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces, Trans. Amer. Math. Soc., 357 (2005), 5001-5029. doi: 10.1090/S0002-9947-05-03638-X.

[8]

R. F. Bass and E. A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc., 355 (2002), 373-405. doi: 10.1090/S0002-9947-02-03120-3.

[9]

H. Brezis, W. Rosenkrants and B. Singer, On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math., 24 (1971), 395-416. doi: 10.1002/cpa.3160240305.

[10]

M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390. doi: 10.1007/s000130050210.

[11]

M. Campiti and I. Rasa, Qualitative properties of a class of Fleming-Viot operators, Acta Math. Hungar., 103 (2004), 55-69. doi: 10.1023/B:AMHU.0000028236.59446.da.

[12]

S. Cerrai and P. Clément, On a class of degenerate elliptic operators arising from the Fleming-Viot processes, J. Evol. Equ., 1 (2001), 243-276. doi: 10.1007/PL00001370.

[13]

S. Cerrai and P. Clément, Schauder estimates for a degenerate second-order elliptic operator on a cube, J. Differential Equations, 242 (2007), 287-321. doi: 10.1016/j.jde.2007.08.002.

[14]

P. Clément and C. A. Timmermans, On $C_0$-semigroup generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math., 89 (1986), 379-387.

[15]

J. R. Dorroh, Contraction semi-groups in a function space, Pacific J. Math., 19 (1966), 35-38. doi: 10.2140/pjm.1966.19.35.

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, Berlin, Heildelberg, 2000.

[17]

C. L. Epstein and R. Mazzeo, Wright-Fisher diffusion in one dimension, SIAM J. Math. Anal., 42 (2010), 1429-1436. doi: 10.1137/090766152.

[18]

C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Annals of Math. Studies, Princeton University Press, 2012.

[19]

S. N. Ethier, A class of degenerate diffusion processes occurring in population genetics, Comm. Pure Appl. Math., 29 (1976), 483-493. doi: 10.1002/cpa.3160290503.

[20]

S. N. Ethier and T. G. Kurtz, Markov Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1986. doi: 10.1002/9780470316658.

[21]

S. N. Ethier and T. G. Kurtz, Fleming-Viot processes in population genetics, SIAM J. Control Optim., 31 (1993), 345-386. doi: 10.1137/0331019.

[22]

W. Feller, Two singular diffusion problems, Ann. of Math., 54 (1951), 173-181. doi: 10.2307/1969318.

[23]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. doi: 10.2307/1969644.

[24]

W. H. Fleming and M. Viot, Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J., 28 (1979), 817-843. doi: 10.1512/iumj.1979.28.28058.

[25]

H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1980.

[26]

G. Köthe, Topological Vector Spaces II, Springer Verlag, Berlin-Heidelberg-New York, 1979.

[27]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[28]

G. Metafune, Analiticity for some degenerate one-dimensional evolution equations, Studia Math., 127 (1998), 251-276.

[29]

R. Nagel, One-Parameter Semigroups of Positive Operators, Lect. Notes Math., 1184, Springer, 1986.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

S. Pal, Analysis of the market weights under the volatility-stabilized market mode, Ann. App. Prob., 21 (2011), 1180-2013. doi: 10.1214/10-AAP725.

[32]

N. Shimakura, Equations différentielles provenant de la génétique des populations, Tôhoku Math. J., 77 (1977), 287-318.

[33]

N. Shimakura, Formulas for diffusion approximations of some gene frequency models, J. Math. Kyoto Univ., 21 (1981), 19-45.

[34]

N. Shimakura, Partial Differential Operators of Elliptic Type, Translations of Mathematical Monographs, 99, Amer. Math. Soc., Providence, 1992.

[35]

W. Stannat, On the validity of the logarithmic-Sobolev inequality for symmetric Fleming-Viot operators, Annals Prob., 28 (2000), 667-684. doi: 10.1214/aop/1019160256.

[36]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.

[1]

Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758

[2]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361

[3]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[4]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[5]

José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195

[6]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[7]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure and Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

[8]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

[9]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477

[10]

Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022

[11]

Piotr Grabowski. On analytic semigroup generators involving Caputo fractional derivative. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022014

[12]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

[13]

Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627

[14]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963

[15]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581

[16]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507

[17]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411

[18]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[19]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[20]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]