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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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.02.042.  Google Scholar

[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.  doi: 10.3934/dcds.2009.23.639.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.01.015.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1090/S0002-9947-05-03638-X.  Google Scholar

[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.  doi: 10.1090/S0002-9947-02-03120-3.  Google Scholar

[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.  doi: 10.1002/cpa.3160240305.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/PL00001370.  Google Scholar

[13]

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

[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.   Google Scholar

[15]

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

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000).   Google Scholar

[17]

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

[18]

C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Annals of Math. Studies, (2012).   Google Scholar

[19]

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

[20]

S. N. Ethier and T. G. Kurtz, Markov Processes,, Wiley Series in Probability and Mathematical Statistics, (1986).  doi: 10.1002/9780470316658.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

H. Jarchow, Locally Convex Spaces,, Teubner, (1980).   Google Scholar

[26]

G. Köthe, Topological Vector Spaces II,, Springer Verlag, (1979).   Google Scholar

[27]

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

[28]

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

[29]

R. Nagel, One-Parameter Semigroups of Positive Operators,, Lect. Notes Math., (1184).   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

N. Shimakura, Partial Differential Operators of Elliptic Type,, Translations of Mathematical Monographs, (1992).   Google Scholar

[35]

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

[36]

F. Treves, Topological Vector Spaces, Distributions and Kernels,, Academic Press, (1967).   Google Scholar

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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.02.042.  Google Scholar

[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.  doi: 10.3934/dcds.2009.23.639.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.01.015.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1090/S0002-9947-05-03638-X.  Google Scholar

[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.  doi: 10.1090/S0002-9947-02-03120-3.  Google Scholar

[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.  doi: 10.1002/cpa.3160240305.  Google Scholar

[10]

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

[11]

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

[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.  doi: 10.1007/PL00001370.  Google Scholar

[13]

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

[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.   Google Scholar

[15]

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

[16]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (2000).   Google Scholar

[17]

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

[18]

C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Annals of Math. Studies, (2012).   Google Scholar

[19]

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

[20]

S. N. Ethier and T. G. Kurtz, Markov Processes,, Wiley Series in Probability and Mathematical Statistics, (1986).  doi: 10.1002/9780470316658.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

H. Jarchow, Locally Convex Spaces,, Teubner, (1980).   Google Scholar

[26]

G. Köthe, Topological Vector Spaces II,, Springer Verlag, (1979).   Google Scholar

[27]

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

[28]

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

[29]

R. Nagel, One-Parameter Semigroups of Positive Operators,, Lect. Notes Math., (1184).   Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

N. Shimakura, Partial Differential Operators of Elliptic Type,, Translations of Mathematical Monographs, (1992).   Google Scholar

[35]

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

[36]

F. Treves, Topological Vector Spaces, Distributions and Kernels,, Academic Press, (1967).   Google Scholar

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