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On the asymptotic stability of Volterra functional equations with vanishing delays
Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift
1. | Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia |
2. | Dipartimento di Matematica E. De Giorgi, Università del Salento, 73100, Lecce |
3. | Dipartimento di Matematica "Ennio De Giorgi”, Università del Salento, C.P. 193, Lecce, I-73100 |
4. | Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe |
References:
[1] |
M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390.
doi: 10.1007/s000130050210. |
[2] |
P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().
|
[3] |
P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965.
doi: 10.1090/S0894-0347-98-00277-X. |
[4] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. |
[5] |
C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Ann. Math. Stud. 185, Princeton Univ. Press, 2013. |
[6] |
P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities,, preprint, ().
|
[7] |
P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations, J. Differential Equations, 256 (2014), 895-956.
doi: 10.1016/j.jde.2013.08.012. |
[8] |
W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. |
[9] |
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 97 (1954), 1-31. |
[10] |
S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary, J. Math. Pures Appl., 87 (2007), 367-393.
doi: 10.1016/j.matpur.2007.02.001. |
[11] |
S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions, J. Differential Equations, 252 (2012), 1182-1212.
doi: 10.1016/j.jde.2011.09.017. |
[12] |
S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces, J. Math. Anal. Appl., 402 (2013), 308-318.
doi: 10.1016/j.jmaa.2013.01.030. |
[13] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().
|
[15] |
J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half-space, Trans. Amer. Math. Soc., 351 (1999), 4611-4628.
doi: 10.1090/S0002-9947-99-02349-1. |
[16] |
K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains, Comm. Partial Differential Equations, 32 (2007), 1261-1280.
doi: 10.1080/03605300600910449. |
[17] |
H. Koch, Non-euclidean singular integrals and the porous medium equation, Habilitation thesis (1999), www.math.uni-bonn.de/simkoch/public.html |
[18] |
J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order, Comm. Pure Appl. Math., 20 (1967), 797-872. |
[19] |
P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, (2004), 65-311.
doi: 10.1007/978-3-540-44653-8_2. |
[20] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, 1995.
doi: 10.1007/978-3-0348-9234-6. |
[21] |
A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa, 2009. |
[22] |
G. Metafune, Analyticity for some degenerate one-dimensional evolution equation, Studia Math., 127 (1998), 251-276. |
[23] |
O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form. Plenum Press, 1973. |
[24] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005. |
[25] |
N. Shimakura, Partial Differential Operators of Elliptic Type, Amer. Math. Soc., Providence (RI), 1992. |
[26] |
K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275-312. |
[27] |
H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations, Marcel Dekker, 1997. |
[28] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. |
show all references
References:
[1] |
M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390.
doi: 10.1007/s000130050210. |
[2] |
P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().
|
[3] |
P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965.
doi: 10.1090/S0894-0347-98-00277-X. |
[4] |
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. |
[5] |
C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Ann. Math. Stud. 185, Princeton Univ. Press, 2013. |
[6] |
P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities,, preprint, ().
|
[7] |
P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations, J. Differential Equations, 256 (2014), 895-956.
doi: 10.1016/j.jde.2013.08.012. |
[8] |
W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. |
[9] |
W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 97 (1954), 1-31. |
[10] |
S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary, J. Math. Pures Appl., 87 (2007), 367-393.
doi: 10.1016/j.matpur.2007.02.001. |
[11] |
S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions, J. Differential Equations, 252 (2012), 1182-1212.
doi: 10.1016/j.jde.2011.09.017. |
[12] |
S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces, J. Math. Anal. Appl., 402 (2013), 308-318.
doi: 10.1016/j.jmaa.2013.01.030. |
[13] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().
|
[15] |
J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half-space, Trans. Amer. Math. Soc., 351 (1999), 4611-4628.
doi: 10.1090/S0002-9947-99-02349-1. |
[16] |
K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains, Comm. Partial Differential Equations, 32 (2007), 1261-1280.
doi: 10.1080/03605300600910449. |
[17] |
H. Koch, Non-euclidean singular integrals and the porous medium equation, Habilitation thesis (1999), www.math.uni-bonn.de/simkoch/public.html |
[18] |
J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order, Comm. Pure Appl. Math., 20 (1967), 797-872. |
[19] |
P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, (2004), 65-311.
doi: 10.1007/978-3-540-44653-8_2. |
[20] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, 1995.
doi: 10.1007/978-3-0348-9234-6. |
[21] |
A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa, 2009. |
[22] |
G. Metafune, Analyticity for some degenerate one-dimensional evolution equation, Studia Math., 127 (1998), 251-276. |
[23] |
O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form. Plenum Press, 1973. |
[24] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005. |
[25] |
N. Shimakura, Partial Differential Operators of Elliptic Type, Amer. Math. Soc., Providence (RI), 1992. |
[26] |
K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275-312. |
[27] |
H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations, Marcel Dekker, 1997. |
[28] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978. |
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