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    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

Angela A. Albanese 1, and  Elisabetta M. Mangino 2,

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.
Keywords: Degenerate elliptic second order operator, space of continuous functions., analytic $C_0$--semigroup, domain with corners.
Mathematics Subject Classification: Primary: 35K65, 35B65, 47D07; Secondary: 60J3.
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.

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