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The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity
Wiener-Landis criterion for Kolmogorov-type operators
| 1. | Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino "Carlo Bo", Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy |
| 2. | Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy |
| 3. | Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5 - 00185 Roma, Italy |
We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
References:
| [1] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
| [2] |
C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158. |
| [3] |
L. C. Evans and R. F. Gariepy,
Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.
doi: 10.1007/BF00249583. |
| [4] |
E. B. Fabes, N. Garofalo and E. Lanconelli,
Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.
doi: 10.1215/S0012-7094-89-05906-1. |
| [5] |
N. Garofalo and E. Lanconelli,
Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.
|
| [6] |
N. Garofalo and E. Lanconelli,
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.
doi: 10.1090/S0002-9947-1990-0998126-5. |
| [7] |
N. Garofalo and F. Segàla,
Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.
doi: 10.1512/iumj.1990.39.39053. |
| [8] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
| [9] |
R. A. Horn and C. R. Johnson,
Matrix Analysis, Cambridge University Press, Cambridge, 1990. |
| [10] |
A.E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
| [11] |
L.P. Kuptsov,
Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.
|
| [12] |
L.P. Kuptsov,
Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.
|
| [13] |
E. Lanconelli,
Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
| [14] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993). |
| [15] |
E. Lanconelli, G. Tralli and F. Uguzzoni,
Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.
doi: 10.1007/s10231-016-0570-y. |
| [16] |
E. Lanconelli and F. Uguzzoni,
Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.
doi: 10.1016/j.jde.2010.01.007. |
| [17] |
E.M. Landis,
Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.
|
| [18] |
M. Manfredini,
The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.
|
| [19] |
V. Scornazzani,
The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.
|
show all references
References:
| [1] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
| [2] |
C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158. |
| [3] |
L. C. Evans and R. F. Gariepy,
Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.
doi: 10.1007/BF00249583. |
| [4] |
E. B. Fabes, N. Garofalo and E. Lanconelli,
Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.
doi: 10.1215/S0012-7094-89-05906-1. |
| [5] |
N. Garofalo and E. Lanconelli,
Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.
|
| [6] |
N. Garofalo and E. Lanconelli,
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.
doi: 10.1090/S0002-9947-1990-0998126-5. |
| [7] |
N. Garofalo and F. Segàla,
Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.
doi: 10.1512/iumj.1990.39.39053. |
| [8] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
| [9] |
R. A. Horn and C. R. Johnson,
Matrix Analysis, Cambridge University Press, Cambridge, 1990. |
| [10] |
A.E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
| [11] |
L.P. Kuptsov,
Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.
|
| [12] |
L.P. Kuptsov,
Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.
|
| [13] |
E. Lanconelli,
Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
| [14] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993). |
| [15] |
E. Lanconelli, G. Tralli and F. Uguzzoni,
Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.
doi: 10.1007/s10231-016-0570-y. |
| [16] |
E. Lanconelli and F. Uguzzoni,
Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.
doi: 10.1016/j.jde.2010.01.007. |
| [17] |
E.M. Landis,
Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.
|
| [18] |
M. Manfredini,
The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.
|
| [19] |
V. Scornazzani,
The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.
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